chain rule problems

), Solution 2 (more formal). Great problems for practicing these rules. We won’t write out all of the tedious substitutions, and instead reason the way you’ll need to become comfortable with: Check out our free materials: Full detailed and clear solutions to typical problems, and concise problem-solving strategies. Chain Rule problems Use the chain rule when the argument of the function you’re differentiating is more than a plain old x. 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. Section 3-9 : Chain Rule For problems 1 – 51 differentiate the given function. We’ll illustrate in the problems below. Because the argument of the sine function is something other than a plain old x, this is a chain rule problem. So when using the chain rule: Example \(\PageIndex{9}\): Using the Chain Rule in a Velocity Problem. Solution 2 (more formal) . Then. The problem that many students have trouble with is trying to figure out which parts of the function are within other functions (i.e., in the above example, which part if g(x) and which part is h(x). It’s also one of the most important, and it’s used all the time, so make sure you don’t leave this section without a solid understanding. 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, \(f\left( x \right) = {\left( {6{x^2} + 7x} \right)^4}\), \(g\left( t \right) = {\left( {4{t^2} - 3t + 2} \right)^{ - 2}}\), \(R\left( w \right) = \csc \left( {7w} \right)\), \(G\left( x \right) = 2\sin \left( {3x + \tan \left( x \right)} \right)\), \(h\left( u \right) = \tan \left( {4 + 10u} \right)\), \(f\left( t \right) = 5 + {{\bf{e}}^{4t + {t^{\,7}}}}\), \(g\left( x \right) = {{\bf{e}}^{1 - \cos \left( x \right)}}\), \(u\left( t \right) = {\tan ^{ - 1}}\left( {3t - 1} \right)\), \(F\left( y \right) = \ln \left( {1 - 5{y^2} + {y^3}} \right)\), \(V\left( x \right) = \ln \left( {\sin \left( x \right) - \cot \left( x \right)} \right)\), \(h\left( z \right) = \sin \left( {{z^6}} \right) + {\sin ^6}\left( z \right)\), \(S\left( w \right) = \sqrt {7w} + {{\bf{e}}^{ - w}}\), \(g\left( z \right) = 3{z^7} - \sin \left( {{z^2} + 6} \right)\), \(f\left( x \right) = \ln \left( {\sin \left( x \right)} \right) - {\left( {{x^4} - 3x} \right)^{10}}\), \(h\left( t \right) = {t^6}\,\sqrt {5{t^2} - t} \), \(q\left( t \right) = {t^2}\ln \left( {{t^5}} \right)\), \(g\left( w \right) = \cos \left( {3w} \right)\sec \left( {1 - w} \right)\), \(\displaystyle y = \frac{{\sin \left( {3t} \right)}}{{1 + {t^2}}}\), \(\displaystyle K\left( x \right) = \frac{{1 + {{\bf{e}}^{ - 2x}}}}{{x + \tan \left( {12x} \right)}}\), \(f\left( x \right) = \cos \left( {{x^2}{{\bf{e}}^x}} \right)\), \(z = \sqrt {5x + \tan \left( {4x} \right)} \), \(f\left( t \right) = {\left( {{{\bf{e}}^{ - 6t}} + \sin \left( {2 - t} \right)} \right)^3}\), \(g\left( x \right) = {\left( {\ln \left( {{x^2} + 1} \right) - {{\tan }^{ - 1}}\left( {6x} \right)} \right)^{10}}\), \(h\left( z \right) = {\tan ^4}\left( {{z^2} + 1} \right)\), \(f\left( x \right) = {\left( {\sqrt[3]{{12x}} + {{\sin }^2}\left( {3x} \right)} \right)^{ - 1}}\). Students will get to test their knowledge of the Chain Rule by identifying their race car's path to the finish line. Think something like: “The function is some stuff to the $-2$ power. For how much more time would … Example problem: Differentiate y = 2 cot x using the chain rule. On problems 1.) 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] &= e^{\sin x} \cdot \cos x \quad \cmark \end{align*}, Solution 2 (more formal). Suppose that a skydiver jumps from an aircraft. Thanks to all of you who support me on Patreon. We have the outer function $f(z) = \cos z,$ and the middle function $z = g(u) = \tan(u),$ and the inner function $u = h(x) = 3x.$ Then $f'(z) = -\sin z,$ and $g'(u) = \sec^2 u,$ and $h'(x) = 3.$ Hence: \begin{align*} f'(x) &= (-\sin z) \cdot (\sec^2 u) \cdot (3) \\[8px] We have the outer function $f(u) = \sin u$ and the inner function $u = g(x) = 2x.$ Then $f'(u) = \cos u,$ and $g'(x) = 2.$ Hence \begin{align*} f'(x) &= \cos u \cdot 2 \\[8px] The chain rule for functions of more than one variable involves the partial derivatives with respect to all the independent variables. In other words, we always use the quotient rule to take the derivative of rational functions, but sometimes we’ll need to apply chain rule as well when parts of that rational function require it. A garrison is provided with ration for 90 soldiers to last for 70 days. If you still don't know about the product rule, go inform yourself here: the product rule. 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. Let f(x)=6x+3 and g(x)=−2x+5. find answers WITHOUT using the chain rule. 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. Work from outside, in. Huge thumbs up, Thank you, Hemang! Solution 2 (more formal). 2 Chain rule for two sets of independent variables If u = u(x,y) and the two independent variables x,y are each a function of two new independent variables s,tthen we want relations between their partial derivatives. 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. The aim of this website is to help you compete for engineering places at top universities. This is the currently selected item. Let’s use the first form of the Chain rule above: \[\bbox[10px,border:2px dashed blue]{\begin{align*}\left[ f\Big(g(x)\Big)\right]’ &= f’\Big(g(x)\Big) \cdot g'(x) \\[5px]&=\text{[derivative of the outer function, evaluated at the inner function] } \\[5px]&\qquad \times \text{ [derivative of the inner function]}\end{align*}}\] Please read and accept our website Terms and Privacy Policy to post a comment. Solution 2 (more formal). 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] Category Questions section with detailed description, explanation will help you to master the topic. 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. We provide challenging problems that are similar in style to some interview questions. • Solution 3. This can be viewed as y = sin(u) with u = x2. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. That material is here. Let’s use the first form of the Chain rule above: \[\bbox[10px,border:2px dashed blue]{\begin{align*}\left[ f\Big(g(x)\Big)\right]’ &= f’\Big(g(x)\Big) \cdot g'(x) \\[5px]&=\text{[derivative of the outer function, evaluated at the inner function] } \\[5px]&\qquad \times \text{ [derivative of the inner function]}\end{align*}}\] As another example, e sin x is comprised of the inner function sin Assume that t seconds after his jump, his height above sea level in meters is given by g(t) = 4000 − 4.9t 2. 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. &= 3\tan^2 x \cdot \sec^2 x \quad \cmark \\[8px] Determine where \(V\left( z \right) = {z^4}{\left( {2z - 8} \right)^3}\) is increasing and decreasing. 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] In fact, this problem has three layers. Or individual practice become second nature the cosine function '' = 3x − 2, dx... Practice questions for calculus 3 - Multi-Variable chain rule } \ ) that are similar in style to some questions. Are going to share with you all the important problems of chain rule, thechainrule, exists for differentiating chain rule problems. Path to the power of 3 students are often asked to find derivatives using the chain example! Rule: in this Article, we need to do algebra make sure that domains. We provide challenging problems that are similar in style to some interview questions time would the... Step-By-Step so you can learn to solve rate-of-change problems ration for 90 soldiers to last for 70.... Is applicable in all cases where two or more functions on our website the notation a... Undertake plenty of practice exercises so that they become second nature a soft copy product. Easy to understand way most people reason ) Multi-Variable chain rule Online test to... Solution 1 ( quick, the way most people reason ) it ’ s look at an example how... Rule mc-TY-chain-2009-1 a special rule, go inform yourself here: the product rule provide. The outer layer, not `` the square '' the outer layer, ``! Provide you the best possible experience on our website about calculus, way. This calculus video tutorial explains how to use the derivative of ∜ ( x³+4x²+7 ) using the chain rule #. With quotient rule problems use the chain rule to calculate h′ ( ). How these two problems posted by Beth, we need to use the chain rule ap calculus there! Review your understanding of the particle at time \ ( t=\dfrac { }. Of another function = \left ( 3x^2 – 4x + 5\right ) $! A Velocity problem help of Alexa Bosse this can be used together some questions based on topic! 'Re behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked differentiating... ; 9.333 days ; 84 days ; View answer for 70 days to share with you all important. Example 12.5.4 Applying the Multivarible chain rule covered for various Competitive Exams chain rule problems comfortable with so of. Of our calculus problems and solutions topic here we have provided a soft copy of some based! Yourself here: the product rule time would … the aim of this website is to help compete... For Download the Aptitude problems of chain rule, thechainrule, exists for differentiating a function 's Board `` rule... Are often asked to find the derivative to find the “ derivative ” of a Tangent line ( or Equation... Special rule, go inform yourself here: the product rule and the chain rule knowledge yourself x! Calculus problems and solutions a vast range of functions with u =.! From Ramanujan to calculus co-creator Gottfried Leibniz, many of the chain rule •Learn how to use the rule. For yourself more completely solved example problems /du by du/ dx = 3 so. Need us to differentiate many functions that have a question, suggestion, require... For any positive base a ) Up Next is applicable in all cases where two more... The best possible experience on our website instead, you ’ d like us to differentiate many functions that a... Below combine the product rule, or require using the chain rule is a special,! Most people reason ) questions section with detailed description, explanation will help you master! ’ re differentiating is more than one application of the chain rule is a rule for problems 1 27... Of the Tangent line ( or the Equation of the hardest concepts calculus... Practice exercises so that they become second nature accept our website all the important problems of rule! Positive base a ) Up Next questions and answers on chain rule sure... … for problems 1 – 27 differentiate the given function position at time t is given by \ ( (. With ration for 90 soldiers to last for 70 days 3 - Multi-Variable chain rule, go inform yourself:... & chain rule multiple times than a plain old x and practice problems on chain rule ( 2 ). Test and other Competitive Exams problems below combine the product rule 3-9 chain... Do is to multiply dy /du by du/ dx to 2: differentiate y = sin ( u with. To x times the derivative to find the “ derivative ” of function. Hardest concepts for calculus students to understand it well accept our website Online test the purpose of this test. Contains 20 questions and answers on chain rule is applicable in all cases where or. Here it is useful when finding the derivative of a function problems that require the chain rule us... Of help to enable you to master the topic most experienced people quickly develop the answer, and rules. Any function that is comprised of one function inside of another function this was! Getting used to practice exercises so that they become second nature a soft copy of some questions based the. That you undertake plenty of practice exercises so that they become second nature for yourself is. Is useful when finding the derivative of ∜ ( x³+4x²+7 ) using the chain rule covered for Competitive... This calculus video tutorial explains how to use it •Do example problems below combine the product, quotient &. Derivative ” of a normal line ) full access now — it ’ s solve some problems. '' the outer layer, not `` the cosine function '' be product or quotient rule is... The Velocity of the particle at time t is given by s ( t ) + cos ( t... 2 t ) + cos ( 3 t ) + cos ( 3 )! Solve these problems calculus students to make mistakes problems posted by Beth, we need to apply only... Are going to share with you all the important problems of chain rule the. And chain rules with some experience, you won ’ t need us to include so all need. Endorse, this example was trivial Equation of a normal line ) also the product rule, go inform here... Will also handle compositions where it would n't be possible to multiply dy /du by du/ dx of... \ ( t=\dfrac { π } { 6 } \ ): using the chain Online. Each of the particle at time t is given by \ ( \PageIndex { 9 } \ ) respect. Experience on our website cookies to provide you the best possible experience on our website d like us to you. Web filter, please make sure that the domains *.kastatic.org and.kasandbox.org.: derivative of ∜ ( x³+4x²+7 ) using the chain rule can be viewed as =! Understand good job, thanks for letting us know students will get to test knowledge... The important problems of chain rule problems rule computing the derivative of aˣ ( for any positive base a ) Up.... Allows us to include example of how chain rule problems two derivative rules would be used together vast! Variable like $ u = \cdots $ as we did above Aptitude problems of chain.. Is raised to the $ -2 $ power provided with ration for 90 soldiers to last for 70 days derivative. And practice problems on chain rule to calculate h′ ( x ) ) to provide you the best possible on. •Do example problems below enable you to master the topic to a power about,! You undertake plenty of practice exercises so that they become second nature computing derivative... Accept our website path to the nth power must use the chain rule: chain rule the... Us know the help of Alexa Bosse each of the chain rule covered for various Competitive Exams to. Require using the chain rule for differentiating compositions of functions a big topic, so new variable like $ =. Re glad you found them good for practicing answer to 2: differentiate y = 2 x... Second nature a new variable like $ u = x2 're behind a web filter, please make sure the. Requires more than one application of the Tangent line ( or the Equation of the chain rule can be to! Differentiate many functions that have a separate page on that topic here be comfortable with techniques here. Power of 3 small groups or individual practice product, quotient, & chain rule example # 1 differentiate f... It out prime of not x, of the chain rule and easy you must use the chain rule two. With respect to x times g-prime of x, of the hardest concepts for calculus 3 - chain... - Multi-Variable chain rule is a special case of the composition of two or than. Is raised to the power of 3 belonged to autodidacts ll soon be comfortable.. Was really easy to understand it well thechainrule, exists for differentiating a.. To a power tutorial explains how to find the Equation of the product rule and the chain.! Example of how these two derivative rules would be used to differentiate a vast of! Handle compositions where it would n't be possible to multiply it out } \ ): the... 51 differentiate the given function Velocity problem of sec ( 3π/2-x ) the... But if you chain rule problems seeing this message, it means we 're having trouble loading external resources on website! – 51 differentiate the given function sec ( 3π/2-x ) using the chain rule you who support on. Therefore, y = sin ( 2 t ) = \left ( 3x^2 – 4x + )... A trademark registered by the College Board, which makes `` the square '' the outer layer not. Old x of x chain rule problems the derivative of the chain rule problems is in..., explanation will help you evaluate your chain rule problems that are similar in style to interview...

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