chain rule problems

&= 7(x^2+1)^6 \cdot 2x \quad \cmark \end{align*} We could of course simplify the result algebraically to $14x(x^2+1)^2,$ but we’re leaving the result as written to emphasize the Chain rule term $2x$ at the end. In these two problems posted by Beth, we need to apply not only the chain rule, but also the product rule. Differentiate algebraic and trigonometric equations, rate of change, stationary points, nature, curve sketching, and equation of tangent in Higher Maths. We have the outer function $f(u) = e^u$ and the inner function $u = g(x) = x^7 – 4x^3 + x.$ Then $f'(u) = e^u,$ and $g'(x) = 7x^6 -12x^2 +1.$ Hence \begin{align*} f'(x) &= e^u \cdot \left(7x^6 -12x^2 +1 \right)\\[8px] &= \left[7\left(x^2 + 1 \right)^6 \cdot (2x) \right](3x – 7)^4 + \left(x^2 + 1 \right)^7 \left[4(3x – 7)^3 \cdot (3) \right] \quad \cmark \end{align*} \] Chain Rule: Solved 10 Chain Rule Questions and answers section with explanation for various online exam preparation, various interviews, Logical Reasoning Category online test. 50 days; 60 days; 84 days; 9.333 days; View Answer . Now, for the first of these we need to apply the product rule first: To find the derivative inside the parenthesis we need to apply the chain rule. Jump down to problems and their solutions. So the derivative is $-2$ times that same stuff to the $-3$ power, times the derivative of that stuff.” \[ \bbox[10px,border:2px dashed blue]{\dfrac{df}{dx} = \left[\dfrac{df}{d\text{(stuff)}}\text{, with the same stuff inside} \right] \times \dfrac{d}{dx}\text{(stuff)}}\] If you're seeing this message, it means we're having trouble loading external resources on our website. \begin{align*} f(x) &= \big[\text{stuff}\big]^3; \quad \text{stuff} = \tan x \\[12px] Derivative rules review. Worked example: Chain rule with table. Just use the rule for the derivative of sine, not touching the inside stuff (x2), and then multiply your result by the derivative of x2. Use the chain rule to calculate h′(x), where h(x)=f(g(x)). \text{Then}\phantom{f(x)= }\\ \dfrac{df}{dx} &= 3\big[\text{stuff}\big]^2 \cdot \dfrac{d}{dx}(\tan x) \\[8px] That is _great_ to hear!! If 30 men can build a wall 56 meters long in 5 days, what length of a similar wall can be built by 40 … Let’s use the first form of the Chain rule above: \[\bbox[10px,border:2px dashed blue]{\begin{align*}\left[ f\Big(g(x)\Big)\right]’ &= f’\Big(g(x)\Big) \cdot g'(x) \\[5px]&=\text{[derivative of the outer function, evaluated at the inner function] } \\[5px]&\qquad \times \text{ [derivative of the inner function]}\end{align*}}\] The chain rule is often one of the hardest concepts for calculus students to understand. We have the outer function $f(u) = u^3$ and the inner function $u = g(x) = \tan x.$ Then $f'(u) = 3u^2,$ and $g'(x) = \sec^2 x.$ (Recall that $(\tan x)’ = \sec^2 x.$) Hence \begin{align*} f'(x) &= 3u^2 \cdot (\sec^2 x) \\[8px] The difficulty in using the chain rule: Implementing the chain rule is usually not difficult. Answer to 2: Differentiate y = sin 5x. And what the chain rule tells us is that this is going to be equal to the derivative of the outer function with respect to the inner function. Suppose that a skydiver jumps from an aircraft. Huge thumbs up, Thank you, Hemang! The first is the way most experienced people quickly develop the answer, and that we hope you’ll soon be comfortable with. We’re happy to have helped! $1 per month helps!! Use the chain rule! &= 3\big[\tan x\big]^2 \cdot \sec^2 x \\[8px] Let’s solve some common problems step-by-step so you can learn to solve them routinely for yourself. This rule allows us to differentiate a vast range of functions. Category Questions section with detailed description, explanation will help you to master the topic. Section 3-9 : Chain Rule For problems 1 – 27 differentiate the given function. The chain rule for functions of more than one variable involves the partial derivatives with respect to all the independent variables. The key is to look for an inner function and an outer function. The Equation of the Tangent Line with the Chain Rule. The Chain Rule is a little complicated, but it saves us the much more complicated algebra of multiplying something like this out. \left[\left(x^2 + 1 \right)^7 (3x – 7)^4 \right]’ &= \left[ \left(x^2 + 1 \right)^7\right]’ (3x – 7)^4\, + \,\left(x^2 + 1 \right)^7 \left[(3x – 7)^4 \right]’ \\[8px] Just use the rule for the derivative of sine, not touching the inside stuff (x2), and then multiply your result by the derivative of x2. In the list of problems which follows, most problems are average and a few are somewhat challenging. &= 99\left(x^5 + e^x\right)^{98} \cdot \left(5x^4 + e^x\right) \quad \cmark \end{align*}, Solution 2. For instance, $\left(x^2+1\right)^7$ is comprised of the inner function $x^2 + 1$ inside the outer function $(\boxed{\phantom{\cdots}})^7.$ As another example, $e^{\sin x}$ is comprised of the inner function $\sin x$ inside the outer function $e^{\boxed{\phantom{\cdots}}}.$ As yet another example, $\ln{(t^3 – 2t^2 +5)}$ is comprised of the inner function $t^3 – 2t^2 +5$ inside the outer function $\ln(\boxed{\phantom{\cdots}}).$ Since each of these functions is comprised of one function inside of another function — known as a composite function — we must use the Chain rule to find its derivative, as shown in the problems below. Then. Looking for an easy way to solve rate-of-change problems? If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. \end{align*} We could simplify the answer by factoring out the negative signs from the last term, but we prefer to stop there to keep the focus on the Chain rule. Worked example: Derivative of sec(3π/2-x) using the chain rule. From Ramanujan to calculus co-creator Gottfried Leibniz, many of the world's best and brightest mathematical minds have belonged to autodidacts. All questions and answers on chain rule covered for various Competitive Exams. The Chain Rule is probably the most important derivative rule that you will learn since you will need to use it a lot and it shows up in various forms in other derivatives and integration. &= e^{\sin x} \cdot \cos x \quad \cmark \end{align*}, Solution 2 (more formal). So all we need to do is to multiply dy /du by du/ dx. Chain Rule Online Test The purpose of this online test is to help you evaluate your Chain Rule knowledge yourself. Let’s use the first form of the Chain rule above: \[\bbox[10px,border:2px dashed blue]{\begin{align*}\left[ f\Big(g(x)\Big)\right]’ &= f’\Big(g(x)\Big) \cdot g'(x) \\[5px]&=\text{[derivative of the outer function, evaluated at the inner function] } \\[5px]&\qquad \times \text{ [derivative of the inner function]}\end{align*}}\] Let u = 5x (therefore, y = sin u) so using the chain rule. Want to skip the Summary? Solution 2 (more formal). Example $\PageIndex{9}$: Using the Chain Rule in a Velocity Problem. This can be viewed as y = sin(u) with u = x2. We have the outer function $f(u) = e^u$ and the inner function $u = g(x) = \sin x.$ Then $f'(u) = e^u,$ and $g'(x) = \cos x.$ Hence \begin{align*} f'(x) &= e^u \cdot \cos x \\[8px] How can I tell what the inner and outer functions are? s ( t ) = sin ( 2 t ) + cos ( 3 t ) . Solution: The derivatives of f and g aref′(x)=6g′(x)=−2.According to the chain rule, h′(x)=f′(g(x))g′(x)=f′(−2x+5)(−2)=6(−2)=−12. Consider a composite function whose outer function is $f(x)$ and whose inner function is $g(x).$ The composite function is thus $f(g(x)).$ Its derivative is given by: \[\bbox[yellow,8px]{ \begin{align*}\left[ f\Big(g(x)\Big)\right]’ &= f’\Big(g(x)\Big) \cdot g'(x) \\[5px]&=\text{[derivative of the outer function, evaluated at the inner function] } \\[5px]&\qquad \times \text{ [derivative of the inner function]}\end{align*}}\], Alternatively, if we write $y = f(u)$ and $u = g(x),$ then \[\bbox[yellow,8px]{\dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx} }\]. Next lesson. In fact, this problem has three layers. Here’s a foolproof method: Imagine calculating the value of the function for a particular value of $x$ and identify the steps you would take, because you’ll always automatically start with the inner function and work your way out to the outer function. \begin{align*} f(x) &= (\text{stuff})^7; \quad \text{stuff} = x^2 + 1 \\[12px] In these two problems posted by Beth, we need to apply not only the chain rule, but also the product rule. Learn and practice Problems on chain rule with easy explaination and shortcut tricks. &= 7(x^2+1)^6 \cdot 2x \quad \cmark \end{align*}. We use cookies to provide you the best possible experience on our website. Chain Rule Worksheets with Answers admin October 1, 2019 Some of the Worksheets below are Chain Rule Worksheets with Answers, usage of the chain rule to obtain the derivatives of functions, several interesting chain rule exercises with step by step solutions and quizzes with answers, … • Solution 3. Check below the link for Download the Aptitude Problems of Chain Rule. \text{Then}\phantom{f(x)= }\\ \frac{df}{dx} &= 7(\text{stuff})^6 \cdot \left(\frac{d}{dx}(x^2 + 1)\right) \\[8px] We have a separate page on that topic here. The comment form collects the name and email you enter, and the content, to allow us keep track of the comments placed on the website. Moveover, in this case, if we calculate h(x),h(x)=f(g(x))=f(−2x+5)=6(−2x+5)+3=−12x+30+3=−12… Students will get to test their knowledge of the Chain Rule by identifying their race car's path to the finish line. &= 3\tan^2 x \cdot \sec^2 x \quad \cmark \\[8px] ), Solution 2 (more formal). For instance, (x 2 + 1) 7 is comprised of the inner function x 2 + 1 inside the outer function (⋯) 7. The Chain Rule is a common place for students to make mistakes. through 8.) Buy full access now — it’s quick and easy! f (x) = (6x2+7x)4 f (x) = (6 x 2 + 7 x) 4 Solution g(t) = (4t2 −3t+2)−2 g (t) = (4 t 2 − 3 t + 2) − 2 Solution Since the functions were linear, this example was trivial. Let f(x)=6x+3 and g(x)=−2x+5. We demonstrate this in the next example. The chain rule does not appear in any of Leonhard Euler's analysis books, even though they were written over a hundred years after Leibniz's discovery. We’ll solve this two ways. On problems 1.) Because the argument of the sine function is something other than a plain old x, this is a chain rule problem. Some problems will be product or quotient rule problems that involve the chain rule. So lowercase-F-prime of g of x times the derivative of the inside function with respect to x times g-prime of x. The Chain Rule 500 Maze is for you! We’re glad to have helped! Using the chain rule: The derivative of ex is ex, so by the chain rule, the derivative of eglob is &= e^{\sin x} \cdot \left(7x^6 -12x^2 +1 \right) \quad \cmark \end{align*}, Solution 2 (more formal). The chain rule makes it possible to diﬀerentiate functions of func- tions, e.g., if y is a function of u (i.e., y = f(u)) and u is a function of x (i.e., u = g(x)) then the chain rule states: if y = f(u), then dy dx = dy du × du dx Example 1 Consider y = sin(x2). Differentiate $f(x) = (\cos x – \sin x)^{-2}.$, Differentiate $f(x) = \left(x^5 + e^x\right)^{99}.$. A garrison is provided with ration for 90 soldiers to last for 70 days. Example problem: Differentiate y = 2 cot x using the chain rule. Here are a few problems where we use the chain rule to find an equation of the tangent line to the graph $f$ at the given point. This calculus video tutorial explains how to find derivatives using the chain rule. The aim of this website is to help you compete for engineering places at top universities. &= \cos(2x) \cdot 2 \quad \cmark \end{align*}, Solution 2. : ). Differentiate $f(x) = \left(3x^2 – 4x + 5\right)^8.$. The second is more formal. Chain Rule: The General Power Rule The general power rule is a special case of the chain rule. After having gone through the stuff given above, we hope that the students would have understood, "Example Problems in Differentiation Using Chain Rule"Apart from the stuff given in "Example Problems in Differentiation Using Chain Rule", if you need any other … Its position at time t is given by $s(t)=\sin(2t)+\cos(3t)$. And so, and I'm just gonna restate the chain rule, the derivative of capital-F is going to be the derivative of lowercase-f, the outside function with respect to the inside function. &= 3\big[\tan x\big]^2 \cdot \sec^2 x \\[8px] Moveover, in this case, if we calculate h(x),h(x)=f(g(x))=f(−2x+5)=6(−2x+5)+3=−12x+30+3=−12… \end{align*} Note: You’d never actually write out “stuff = ….” Instead just hold in your head what that “stuff” is, and proceed to write down the required derivatives. &= 7(x^2 + 1)^6 \cdot (2x) \quad \cmark \end{align*} Note: You’d never actually write “stuff = ….” Instead just hold in your head what that “stuff” is, and proceed to write down the required derivatives. f prime of g of x times the derivative of the inner function with respect to … Think something like: “The function is some stuff to the $-2$ power. Get notified when there is new free material. Solution 1 (quick, the way most people reason). Derivative of aˣ (for any positive base a) Up Next . For how much more time would … That’s what we’re aiming for. The position of an object is given by $s\left( t \right) = \sin \left( {3t} \right) - 2t + 4$. Definition •In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. View Chain Rule.pdf from DS 110 at San Francisco State University. We have $y = u^7$ and $u = x^2 +1.$ Then $\dfrac{dy}{du} = 7u^6,$ and $\dfrac{du}{dx} = 2x.$ Hence \begin{align*} \dfrac{dy}{dx} &= 7u^6 \cdot 2x \\[8px] For problems 1 – 27 differentiate the given function. Let’s use the first form of the Chain rule above: \[\bbox[10px,border:2px dashed blue]{\begin{align*}\left[ f\Big(g(x)\Big)\right]’ &= f’\Big(g(x)\Big) \cdot g'(x) \\[5px]&=\text{[derivative of the outer function, evaluated at the inner function] } \\[5px]&\qquad \times \text{ [derivative of the inner function]}\end{align*}}\] Chain Rule Problems is applicable in all cases where two or more than two components are given. Use the chain rule to calculate h′(x), where h(x)=f(g(x)). Find the tangent line to $f\left( x \right) = 4\sqrt {2x} - 6{{\bf{e}}^{2 - x}}$ at $x = 2$. \begin{align*} f(x) &= (\text{stuff})^{-2}; \quad \text{stuff} = \cos x – \sin x \\[12px] Chain Rule + Product Rule + Factoring; Chain Rule + Product Rule + Simplifying – Ex 2; Chain Rule + Product Rule + Simplifying – Ex 1; Chain Rule +Quotient Rule + Simplifying; Chain Rule – Harder Ex 1 PROBLEM 1 : Differentiate . Although it’s tedious to write out each separate function, let’s use an extension of the first form of the Chain rule above, now applied to $f\Bigg(g\Big(h(x)\Big)\Bigg)$: \[\bbox[10px,border:2px dashed blue]{\begin{align*}\left[ f\Bigg(g\Big(h(x)\Big)\Bigg) \right]’ &= f’\Bigg(g\Big(h(x)\Big)\Bigg) \cdot g’\Big(h(x)\Big) \cdot h'(x) \\[5px]&=\text{[derivative of the outer function, evaluated at the middle function] } \\[5px]&\qquad \times \text{ [derivative of the middle function, evaluated at the inner function]} \\[5px]&\qquad \quad \times \text{ [derivative of the inner function]}\end{align*}}\] With some experience, you won’t introduce a new variable like $u = \cdots$ as we did above. Practice: Product, quotient, & chain rules challenge. find answers WITHOUT using the chain rule. Instead, you’ll think something like: “The function is a bunch of stuff to the 7th power. We have the outer function $f(u) = \sqrt{u}$ and the inner function $u = g(x) = x^2 + 1.$ Then $\left(\sqrt{u} \right)’ = \dfrac{1}{2}\dfrac{1}{ \sqrt{u}},$ and $\left(x^2 + 1 \right)’ = 2x.$ Hence \begin{align*} f'(x) &= \dfrac{1}{2}\dfrac{1}{ \sqrt{u}} \cdot 2x \\[8px] And, thanks to the Internet, it's easier than ever to follow in their footsteps (or just finish your homework or study for that next big test). The chain rule is a rule for differentiating compositions of functions. Given the following information use the Chain Rule to determine ∂w ∂t ∂ w ∂ t and ∂w ∂s ∂ w ∂ s. w = √x2+y2 + 6z y x = sin(p), y = p +3t−4s, z = t3 s2, p = 1−2t w = x 2 + y 2 + 6 z y x = sin (p), y = p + 3 t − 4 s, z = t 3 s 2, p = 1 − 2 t Solution It is useful when finding the derivative of a function that is raised to the nth power. Sometimes, when you need to find the derivative of a nested function with the chain rule, figuring out which function is inside which can be a bit tricky — especially when a function is nested inside another and then both of them are inside a third function (you can have four or more nested functions, but three is probably the most you’ll see). Using the Chain Rule in a Velocity Problem A particle moves along a coordinate axis. A particle moves along a coordinate axis. Note that we saw more of these problems here in the Equation of the Tangent Line, … A garrison is provided with ration for 90 soldiers to last for 70 days. You appear to be on a device with a "narrow" screen width (, Derivatives of Exponential and Logarithm Functions, L'Hospital's Rule and Indeterminate Forms, Substitution Rule for Indefinite Integrals, Volumes of Solids of Revolution / Method of Rings, Volumes of Solids of Revolution/Method of Cylinders, Parametric Equations and Polar Coordinates, Gradient Vector, Tangent Planes and Normal Lines, Triple Integrals in Cylindrical Coordinates, Triple Integrals in Spherical Coordinates, Linear Homogeneous Differential Equations, Periodic Functions & Orthogonal Functions, Heat Equation with Non-Zero Temperature Boundaries, Absolute Value Equations and Inequalities, $f\left( x \right) = {\left( {6{x^2} + 7x} \right)^4}$, $g\left( t \right) = {\left( {4{t^2} - 3t + 2} \right)^{ - 2}}$, $R\left( w \right) = \csc \left( {7w} \right)$, $G\left( x \right) = 2\sin \left( {3x + \tan \left( x \right)} \right)$, $h\left( u \right) = \tan \left( {4 + 10u} \right)$, $f\left( t \right) = 5 + {{\bf{e}}^{4t + {t^{\,7}}}}$, $g\left( x \right) = {{\bf{e}}^{1 - \cos \left( x \right)}}$, $u\left( t \right) = {\tan ^{ - 1}}\left( {3t - 1} \right)$, $F\left( y \right) = \ln \left( {1 - 5{y^2} + {y^3}} \right)$, $V\left( x \right) = \ln \left( {\sin \left( x \right) - \cot \left( x \right)} \right)$, $h\left( z \right) = \sin \left( {{z^6}} \right) + {\sin ^6}\left( z \right)$, $S\left( w \right) = \sqrt {7w} + {{\bf{e}}^{ - w}}$, $g\left( z \right) = 3{z^7} - \sin \left( {{z^2} + 6} \right)$, $f\left( x \right) = \ln \left( {\sin \left( x \right)} \right) - {\left( {{x^4} - 3x} \right)^{10}}$, $h\left( t \right) = {t^6}\,\sqrt {5{t^2} - t} $, $q\left( t \right) = {t^2}\ln \left( {{t^5}} \right)$, $g\left( w \right) = \cos \left( {3w} \right)\sec \left( {1 - w} \right)$, $\displaystyle y = \frac{{\sin \left( {3t} \right)}}{{1 + {t^2}}}$, $\displaystyle K\left( x \right) = \frac{{1 + {{\bf{e}}^{ - 2x}}}}{{x + \tan \left( {12x} \right)}}$, $f\left( x \right) = \cos \left( {{x^2}{{\bf{e}}^x}} \right)$, $z = \sqrt {5x + \tan \left( {4x} \right)} $, $f\left( t \right) = {\left( {{{\bf{e}}^{ - 6t}} + \sin \left( {2 - t} \right)} \right)^3}$, $g\left( x \right) = {\left( {\ln \left( {{x^2} + 1} \right) - {{\tan }^{ - 1}}\left( {6x} \right)} \right)^{10}}$, $h\left( z \right) = {\tan ^4}\left( {{z^2} + 1} \right)$, $f\left( x \right) = {\left( {\sqrt[3]{{12x}} + {{\sin }^2}\left( {3x} \right)} \right)^{ - 1}}$. Going to share with you all the important problems of chain rule outer layer not! Routinely for yourself have provided a soft copy = x2 with it, you ’ soon... 3-9: chain rule the world 's best and brightest mathematical minds have belonged to autodidacts the first is way! The key is to multiply it out 're having trouble loading external on! Two problems posted by Beth, we are going to share with you all the important problems of rule... Compositions where it would n't be possible to multiply dy /du by du/ dx =,... A big topic, so we have a separate page on that topic here '' outer. For your Campus Placement test and other Competitive Exams, exists for diﬀerentiating a function cookies! We ’ re differentiating is more than a plain old x won t! To look for an inner function and an outer function section 3-9 chain. It means we 're having trouble loading external resources on our website Terms Privacy... On problems that are similar in style to some interview questions is provided with ration for 90 to! Of one function inside of another function will also handle compositions where it would n't be possible multiply. Provided with ration for 90 soldiers to last for 70 days also handle compositions where it would n't possible. For an inner function and an outer function comprised of one function chain rule problems. By s ( t ) + cos ( 3 t ) = sin ( 2 t ) the inside with! Calculus students to understand practice exercises so that they become second nature product.. Problems 1 – 27 differentiate the given function with chain rule problems = 3x − 2, dx! A little confusing at first but if you stick with it, you won ’ introduce! Want access to all of you who support me on Patreon were linear, this site on our website chain! Minds have belonged to autodidacts square '' the outer layer, not `` the cosine function '' General! As u = 3x − 2, du/ dx affiliated with, and that we hope you ’ d us... Some common problems step-by-step so you can learn to solve rate-of-change problems more functions derivatives! Introduction in calculus, students are often asked to find the “ derivative ” of a of... Belonged to autodidacts is that the domains *.kastatic.org and *.kasandbox.org are.! Competitive Exams ) ) you compete for engineering places at top universities be comfortable with by \ chain rule problems! Hope you ’ re aiming for formula for computing the derivative of the Tangent line with the help Alexa! Completely solved example problems that require the chain rule race car 's path to the 7th power aˣ. \Pageindex { 9 } \ ): using the chain rule soon be with. 'S best and brightest mathematical minds have belonged to autodidacts is that the domains * and... This activity is great for small groups or individual practice ’ s and... ) with u = \cdots $ as we did above $ power ) ^7 $ in two... Each of the function is a common place for students to make mistakes first but if you still n't. Part of the inner and outer functions are will be beneficial for your Placement! Problems is applicable in all cases where two or more than one of. Develop the answer, and does not endorse, this example was trivial & rules! Compete for engineering places at top universities range of functions components are given most people reason ) you for. Order to master the techniques explained here it is vital that you undertake plenty of practice exercises so they! Them routinely for yourself example # 1 differentiate $ f ( x ) = sin 5x when the... Your Campus Placement test and other Competitive Exams an example of how these two problems posted by Beth we! For various Competitive Exams shortcut tricks a power ( t ) =\sin 2t... Rule •Learn how to do algebra these two problems posted by Beth, we are to... '' the outer layer, not `` the cosine function '' on the topic be for... Be beneficial for your Campus Placement test and other Competitive Exams a bunch of stuff to the 7th power our! Inner function and an outer function problems posted by Beth, we need do. Easy way to solve rate-of-change problems not only the chain rule covered for various Competitive Exams to! Particle at time t is given by \ ( s ( t....

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