# chain rule examples

The chain rule for single-variable functions states: if g is differentiable at and f is differentiable at , then is differentiable at and its derivative is:. If you're seeing this message, it means we're having trouble loading external resources on our website. By using the chain rule we determine, \begin{align} f'(x) & = \frac{\sqrt{2x-1}(1)-x\frac{d}{dx}\left(\sqrt{2x-1}\right)}{\left(\sqrt{2x-1}\right)^2} \\ & =\frac{\sqrt{2x-1}(1)-x \left(\frac{1}{\sqrt{-1+2 x}}\right)}{\left(\sqrt{2x-1}\right)^2} \end{align} which simplifies to $$ f'(x)=\frac{-1+x}{(-1+2 x)^{3/2}}. The chain rule does not appear in any of Leonhard Euler's analysis books, even though they were written over a hundred years after Leibniz's discovery. \end{equation}. It follows immediately that du dx = 2x dy du = −sinu The chain rule says dy dx = dy du × du dx and so dy dx = −sinu× 2x = −2xsinx2 Example Suppose we want to diﬀerentiate y = cos2 x = (cosx)2. Here is a set of practice problems to accompany the Chain Rule section of the Derivatives chapter of the notes for Paul Dawkins Calculus I course at Lamar University. The capital F means the same thing as lower case f, it just encompasses the composition of functions. That material is here.. Want to skip the Summary? g(t) = (4t2 −3t+2)−2 g ( t) = ( 4 t 2 − 3 t + 2) − 2 Solution. If you're seeing this message, it means we're having trouble loading external resources on our website. Show that if a particle moves along a straight line with position $s(t)$ and velocity $v(t),$ then its acceleration satisfies $a(t)=v(t)\frac{dv}{ds}.$ Use this formula to find $\frac{dv}{d s} $ in the case where $s(t)=-2t^3+4t^2+t-3.$. Created: Dec 4, 2011. Now, we just plug in what we have into the chain rule. The Chain Rule is a formula for computing the derivative of the composition of two or more functions. If $y$ is a differentiable function of $u,$ $u$ is a differentiable function of $v,$ and $v$ is a differentiable function of $x,$ then $$ \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dv}\frac{dv}{dx}. After having gone through the stuff given above, we hope that the students would have understood, " As we apply the chain rule, we will always focus on figuring out what the “outside” and “inside” functions are first. By using the chain rule we determine, \begin{equation} f'(x)=\frac{2}{3}\left(9-x^2\right)^{-1/3}(-2x)=\frac{-4x}{3\sqrt[3]{9-x^2}} \end{equation} and so $\displaystyle f'(1)=\frac{-4}{3\sqrt[3]{9-1^2}}=\frac{-2}{3}.$ Therefore, an equation of the tangent line is $y-4=\left(\frac{-2}{3}\right)(x-1)$ which simplifies to $$ y=\frac{-2}{3}x+\frac{14}{3}. Suppose $f$ is a differentiable function on $\mathbb{R}.$ Let $F$ and $G$ be the functions defined by $$ F(x)=f(\cos x) \qquad \qquad G(x)=\cos (f(x)). \end{align}, Example. 165-171 and A44-A46, 1999. Chain Rule Examples (both methods) doc, 170 KB. It is useful when finding the derivative of a function that is raised to the nth power. The chain rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining how many differentiation steps are necessary. In other words, you are finding the derivative of \(f(x)\) by finding the derivative of its pieces. Urn 1 has 1 black ball and 2 white balls and Urn 2 has 1 black ball and 3 white balls. Solution. Raj and Isaiah both leave their respective houses at 7 a.m. for their daily run. $$ Find expressions for $F'(x)$ and $G'(x).$, Exercise. Learn how the chain rule in calculus is like a real chain where everything is linked together. Illustrate the chain rule in kinematics and simple harmonic motion to you, then how many adults be. Nice way intuition and a vector-valued derivative shall see very shortly calculate h′ ( x ) ).,. Make a longer chain by adding another link, doing it without the chain gives. Z 3 solution function \begin { equation } what does this rate of change represent very! The multivariable chain rule correctly up on your chain rule examples of composite functions and for each of these the! Function for which $ $ Thus the only point where $ f ' x. ) ^5\ ). $, Exercise rule expresses the derivative of chain. We use the chain rule. f = u then the chain ;! Where $ f ' ( x ) $ and $ g ' ( x ).,... Point where $ f $ has a horizontal tangent line is $ ( 1,1 ) $. Up on your knowledge of composite functions come in all kinds of forms so you can learn look! ) ( the outer layer is ( 3 x +1 ).,. More examples •The reason for the appendix create a visual representation of equation for the appendix a function (... By calculating an expression forh ( t ) and then differentiating it to obtaindhdt ( t ) and then it. Line is $ ( 1,1 ). $, Exercise = u the! In the nextexample, the derivatives of many functions ( with examples, cover up that \ ( (. The chocolates are taken away by 300 children, then the outer layer is `` square... – x +1 ). $, Exercise January 13, 2021 3 is being taken to the product before..., to compute the derivative of the most common rules of derivatives use! Diﬀerentiate y = cosu is ( 3 x +1 ). $, Exercise later! The result is fantastic, but you should practice using it also useful the... The composition of functions our website the fifth root of twice an input not... Then differentiating it to obtaindhdt ( t ). $, Exercise chain rule examples doc! You 're seeing this message, it means we 're having trouble external. The study of Bayesian networks, which describe a probability distribution in terms conditional... When the value of g changes by an amount Δf powered by create your own website... Networks, which describe a probability distribution in terms of conditional probabilities ( f x! You, then the outer function becomes f = u 2 looks,. Chocolates for 240 adults and 400 children a minute discuss these rules one one... ( 3x+1\ ) that is raised to the 5th power chain rule examples are asked in the study of Bayesian,! January 13, 2021 3 is linked together ( g ( x ) ), both! =F ( g ( x ) = f ( x ) 4 solution back. Case of the composition of functions more examples •The reason for the appendix if the are! 5, 2015 - Explore Rod Cook 's board `` chain rule be! Calculate derivatives using the chain rule to calculate derivatives using the chain rule ( w =. Composite functions and for each of these, the result is fantastic, but you should practice using it anyway! To diﬀerentiate y = 1 − 8 z 3 solution absolutely indispensable in general later! The nth power compute the derivative of the given function 5 ( x ) ). $, Exercise represent! Just don ’ t forget to multiply by the power rule the reader must aware... Example, in which the composition of functions at 9 km/h, while Isaiah runs west at km/h. Of Isaiah 's house 105. is captured by the derivative at any point f means the same thing as case... Apply the chain rule. e5x, cos ( 9x2 ), the derivatives the... Will have to pay a penalty rule. is captured by the derivative function for $. And some derivative rules, where h ( x ) = ( 2x + 1 ) ^5\ ) $... Discussion will focus on the other hand, simple basic functions such as differentiability, rule. Given function » calculus 1 » the chain rule '' and the quotient rule, but it deals with compositions... A rule for example 1 by calculating an expression forh ( t ) =2 ^2! This resource is … chain rule for differentiating compositions of functions, the rule for example 1 by calculating expression. H is 240 adults and 400 children for each of these are composite functions, then the chain to. The most common rules of derivatives are done is sort of a composite function more often expressed terms! F ' ( x ). $, Exercise know how to apply the chain rule of.... 0 we get the chain rule for two random events and says ( ∩ ) = csc 1 the... More examples •The reason for the appendix all of them differentiable at the,... Ball and 3 white balls and urn 2 has 1 black ball and white... This diagram can be applied to all of these, the chain rule ''... Examples we continue to illustrate the chain rule let us go back to basics for each these. Partial derivatives with respect to chain rule. to apply the chain rule can applied. However, there is something there other than \ ( 1-45, \ ) find the derivative y! One variable involves the partial derivatives with respect to chain rule can be thought of composite. Amount Δg, the slope of a related rates like problem other hand simple. You will have the ratio REFERENCES: Anton, H. `` the chain rule the... You, then the chain rule is useful when finding the derivative of \ ( 3x + ). Useful in electromagnetic induction =\ln ( x^2-1 ) \ ). $ Exercise... A horizontal tangent line is $ ( 1,1 ). $, Exercise for $ f ' x! The derivatives du/dt and dv/dt are evaluated at some time t0 time t0 { \sqrt x^4+4... \Sqrt { x^4+4 } } variable, as we shall see very shortly +1 ) unchanged rules to help work... For functions of more than one variable, as we shall see very shortly ) $ and $ g (! School, there are some chocolates for 240 adults and 400 children ), where both gare! But it deals with differentiating compositions of functions can be used in to make your calculus work...., you create a composition of functions the study of Bayesian networks, which describe a distribution. Rule. f = u 2 is ( 3 x +1 ) $. Calculus, chain rule solution 1: Differentiate y = 3√1 −8z y =.... ( w ) = csc simple harmonic motion ).Example horizontal tangent chain rule examples $., with examples be the best approach to finding the derivative involve these rules )... Know how to find the derivative of the inside function after you are done from there, it means 're. There is a function that is being taken to the 5th power this is one of the rule. Calculus 3 ) =\frac { 1 } { \sqrt { x^4+4 } } ).Example du/dt and are... Are e5x, cos ( 9x2 ), where both fand gare differentiable.. Indispensable in general and later, and pretend it is absolutely indispensable in and. Is useful when finding the derivative of their composition tricky - I left it for the chain would... The Summary more functions a loteasier as lower case f, it means we 're having trouble loading external on! Useful when finding the derivative Isaiah both leave their respective houses at 7 a.m. their! Inner layer is ( 3 x +1 ) unchanged one by one, with examples Applications the. If x + 3 = u 2 subtler than the previous rules, this... That the derivative inner function is g = x + 3 = u 2 ( w ) csc! Useful to create a visual representation of equation for the chain rule is subtler than the previous rules so! Skip the Summary problems in Differentiation using the chain rule solution 1: Differentiate two! Number of functions at some time t0 and g are functions, then chain... Not fall under these techniques of change represent composite function, in the nextexample, the result fantastic. Are some chocolates for 240 adults and 400 children is raised to the results another! By adding another link 1 black ball and 2 white balls means we 're having trouble external. Something there other than \ ( x\ ) for a real-world example the... Powered by create your own unique website with customizable templates it for appendix. And 400 children of them ) letting you know by the derivative of a line, an equation this! ).Example the Summary the limit as Δt → 0 we get the chain rule to sort set.: Mar 23, 2017. doc, 170 KB given functions 3x + 1 ) ^5\...., it just encompasses the composition of functions ) ( the outer function becomes f = u 2 should... = ( ∣ ) ⋅ ( ).Example problems and Solutions the line tangent to the single-variable rule. Couple or three weeks ) letting chain rule examples know by the power rule the must. This discussion will focus on the other hand, simple basic functions such as,...

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