Solution. Then the previous expression is equal to the product of two factors: Next lesson. Example. The chain rule is used to differentiate composite functions. When u = u(x,y), for guidance in working out the chain rule, write down the differential δu= ∂u ∂x δx+ ∂u ∂y δy+ ... (3) then when x= x(s,t) and y= y(s,t) (which are known functions of sand t), the … Here is the chain rule again, still in the prime notation of Lagrange. If $y$ is a differentiable function of $u,$ $u$ is a differentiable function of $v,$ and $v$ is a differentiable function of $x,$ then $$ \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dv}\frac{dv}{dx}. Proving the chain rule. Find the derivative of the following functions.$(1) \quad \displaystyle r=-(\sec \theta +\tan \theta )^{-1}$$(2) \quad \displaystyle y=\frac{1}{x}\sin ^{-5}x-\frac{x}{3}\cos ^3x$$(3) \quad \displaystyle y=(4x+3)^4(x+1)^{-3}$$(4) \quad \displaystyle y=(1+2x)e^{-2x}$$(5) \quad \displaystyle h(x)=x \tan \left(2 \sqrt{x}\right)+7$$(6) \quad \displaystyle g(t)=\left(\frac{1+\cos t}{\sin t}\right)^{-1}$$(7) \quad \displaystyle q=\sin \left(\frac{t}{\sqrt{t+1}}\right)$$(8) \quad \displaystyle y=\theta ^3e^{-2\theta }\cos 5\theta $$(9) \quad \displaystyle y=(1+\cos 2t)^{-4}$$(10) \quad \displaystyle y=\left(e^{\sin (t/2)}\right)^3$$(11) \quad \displaystyle y=\left(1+\tan ^4\left(\frac{t}{12}\right)\right)^3$$(12) \quad \displaystyle y=4 \sin \left(\sqrt{1+\sqrt{t}}\right)$$(13) \quad \displaystyle y=\frac{1}{9}\cot (3x-1)$$(14) \quad \displaystyle y=\sin \left(x^2e^x\right)$$(15) \quad \displaystyle y=e^x \sin \left(x^2e^x\right)$, Exercise. Theorem 1 (Chain Rule). If Δx is an increment in x and Δu and Δy are the corresponding increment in u and y, then we can use Equation(1) to write Δu = g’(a) Δx + ε 1 Δx = * g’(a) + ε A Quotient Rule is stated as the ratio of the quantity of the denominator times the derivative of the numerator function … \end{equation} Thus, \begin{equation} \frac{dv}{d s}=\frac{-12t+8}{-6t^2+8t+1}. 5 Stars: 5: … The following is a proof of the multi-variable Chain Rule. Example. Under certain conditions, such as differentiability, the result is fantastic, but you should practice using it. \end{align}, Example. Given $y=6u-9$ and find $\frac{dy}{dx}$ for (a) $u=(1/2)x^4$, (b) $u=-x/3$, and (c) $u=10x-5.$, Exercise. Show that if a particle moves along a straight line with position $s(t)$ and velocity $v(t),$ then its acceleration satisfies $a(t)=v(t)\frac{dv}{ds}.$ Use this formula to find $\frac{dv}{d s} $ in the case where $s(t)=-2t^3+4t^2+t-3.$. Proof. Translating the chain rule into Leibniz notation. Dave will teach you what you need to know. f [ g ( x)] – f [ g ( c)] x – c = Q [ g ( x)] g ( x) − g ( c) x − c. for all x in a punctured neighborhood of c. In which case, the proof of Chain Rule can be finalized in a few steps through the use of limit laws. For each of the following functions, write the function ${y=f(x)}$ in the form $y=f(u)$ and $u=g(x)$, then find $\frac{dy}{dx}.$$(1) \quad \displaystyle y=\left(\frac{x^2}{8}+x-\frac{1}{x}\right)^4$$(2) \quad \displaystyle y=\sec (\tan x)$$(3) \quad \displaystyle y=5 \cos ^{-4}x$$(4) \quad \displaystyle y=e^{5-7x}$ $(5) \quad \displaystyle y=\sqrt{2x-x^2}$$(6) \quad \displaystyle y=e^x \sqrt{2x-x^2}$, Exercise. Copyright © 2020 Dave4Math LLC. This proof uses the following fact: Assume , and . ), Calculus (Start Here) – Enter the World of Calculus, Continuous (It’s Meaning and Applications), Derivative Definition (The Derivative as a Function), Derivative Examples (The Role of the Derivative), Find the Limit (Techniques for Finding Limits), First Derivative Test (and Curve Sketching), Horizontal Asymptotes and Vertical Asymptotes, Implicit Differentiation (and Logarithmic Differentiation), L ‘Hopital’s Rule and Indeterminate Forms, Limit Definition (Precise Definition of Limit), Choose your video style (lightboard, screencast, or markerboard). However, we can get a better feel for it using some intuition and a couple of examples. The lecture Chain Rule Proof by Batool Akmal is from the course Quotient Rule, Chain Rule and Product Rule. It's a "rigorized" version of the intuitive argument given above. dy/dx = F'(H(x)).H'(x) dy/dx = F'(H(x)).H(x) dy/dx = F'(H(x)) dy/dx = F'(H(x)) / H'(x) dy/dx = F'(H(x)) + H'(x) Author of lecture Chain Rule Proof. Batool Akmal. 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 $$ f'(x)=\frac{-1+x}{(-1+2 x)^{3/2}}. Given a2R and functions fand gsuch that gis differentiable at aand fis differentiable at g(a). \end{equation} as desired. Solution. Solution. It follows that f0[g(x)] = lim ∆g→0 f[g(x)+∆g]−f[g(x)] ∆g = lim ∆x→0 f[g(x+∆x)]−f[g(x)] g(x+∆x)−g(x) = lim ∆x→0 %PDF-1.4 As in single variable calculus, there is a multivariable chain rule. If y = (1 + x²)³ , find dy/dx . Derivative rules review. One proof of the chain rule begins with the definition of the derivative: (∘) ′ = → (()) − (()) −. Theorem. To begin with, let us introduce a variable u = g(x) to simplify the looks of our steps. =_.���tK���L���d�&-.Y�Y&M6���)j-9Ә��cA�a�h,��4���2�e�He���9Ƶ�+nO���^b��j�(���{� Suppose that $u=g(x)$ is differentiable at $x=-5,$ $y=f(u)$ is differentiable at $u=g(-5),$ and $(f\circ g)'(-5)$ is negative. Example. It is especially transparent using o() notation, where once again f(x) = o(g(x)) means that lim x!0 f(x) g(x) = 0: Then justify your claim. Worked example: Derivative of sec(3π/2-x) using the chain rule. \end{equation} Therefore, \begin{equation} g'(2)=2(2) f\left(\frac{2}{2-1}\right)+2^2f’\left(\frac{2}{2-1}\right)\left(\frac{-1}{(2-1)^2}\right)=-24. h→0. Find the derivative of the function \begin{equation} g(x)=\left(\frac{3x^2-2}{2x+3}\right)^3. \end{align} as needed. \end{equation}. �Vq ���N�k?H���Z��^y�l6PpYk4ږ�����=_^�>�F�Jh����n� �碲O�_�?�W�Z��j"�793^�_=�����W��������b>���{� =����aޚ(�7.\��� l�����毉t�9ɕ�n"�� ͬ���ny�m�`�M+��eIǬѭ���n����t9+���l�����]��v���hΌ��Ji6I�Y)H\���f $(1) \quad \displaystyle g(\Delta u)=\frac{f[u(x)+\Delta u]-f[u(x)]}{\Delta u}-\frac{df}{du}$ provided $\Delta u\neq 0$ $(2) \quad \displaystyle \left[g(\Delta u)+\frac{df}{du}\right]\Delta u=f[u(x)+\Delta u]-f[u(x)]$ $(3) \quad g$ is continuous at $t=0$ since $$ \lim_{t\to 0} \left[ \frac{f[u(x)+t]-f[u(x)]}{t}\right]=\frac{df}{du}. By the chain rule, \begin{equation} a(t)=\frac{dv}{dt}=\frac{dv}{d s}\frac{ds}{dt}=v(t)\frac{dv}{ds} \end{equation} In the case where $s(t)=-2t^3+4t^2+t-3; $ we determine, \begin{equation} \frac{ds}{dt} = v(t) = -6t^2+8t+1 \qquad \text{and } \qquad a(t)=-12t+8. All rights reserved. /Filter /FlateDecode Define \(\phi = f\circ \mathbf g\). \end{equation} What does this rate of change represent? and M.S. Implicit differentiation. Then using the definition of the derivative, we can write u'(x) as: u′(x)=Limx→0u(x+h)–u(x)h{u'(x) = Lim_{x\rightarrow{0}}\frac{u(x + h) – u(x)}{h}} u′(x)=Limx→0​hu(x+h)–… Suppose that the functions $f$, $g$, and their derivatives with respect to $x$ have the following values at $x=0$ and $x=1.$ \begin{equation} \begin{array}{c|cccc} x & f(x) & g(x) & f'(x) & g'(x) \\ \hline 0 & 1 & 1 & 5 & 1/3 \\ 1 & 3 & -4 & -1/3 & -8/3 \end{array} \end{equation} Find the derivatives with respect to $x$ of the following combinations at a given value of $x,$ $(1) \quad \displaystyle 5 f(x)-g(x), x=1$ $(2) \quad \displaystyle f(x)g^3(x), x=0$ $(3) \quad \displaystyle \frac{f(x)}{g(x)+1}, x=1$$(4) \quad \displaystyle f(g(x)), x=0$ $(5) \quad \displaystyle g(f(x)), x=0$ $(6) \quad \displaystyle \left(x^{11}+f(x)\right)^{-2}, x=1$$(7) \quad \displaystyle f(x+g(x)), x=0$$(8) \quad \displaystyle f(x g(x)), x=0$$(9) \quad \displaystyle f^3(x)g(x), x=0$. The version with several variables is more complicated and we will use the tangent approximation and total differentials to help understand and organize it. Suppose that f is differentiable at the point \(\displaystyle P(x_0,y_0),\) where \(\displaystyle x_0=g(t_0)\) and … Example. It is used where the function is within another function. Assuming that the following derivatives exists, find \begin{equation} \frac{d}{d x}f’ [f(x)] \qquad \text{and}\qquad \frac{d}{d x}f [f'(x)]. You can use our resources like sample question papers and Maths previous years’ papers to practise questions and answers for Maths board exam preparation. Using the differentiation rule $\frac{d}{dx}[\ln u]=\frac{u’}{u};$ we have, \begin{equation} \frac{d}{d x}( \ln |\cos x| ) =\frac{1}{\cos x}\frac{d}{dx}(\cos x) =\frac{\sin x}{\cos x} =\tan x \end{equation} and \begin{align} & \frac{d}{d x}( (\ln |\sec x+\tan x|) ) \\ & \qquad =\frac{1}{|\sec x+\tan x|}\frac{d}{dx}(|\sec x+\tan x|) \\ & \qquad = \frac{1}{|\sec x+\tan x|}\frac{\sec x+\tan x}{|\sec x+\tan x|}\frac{d}{dx}(\sec x+\tan x) \\ & \qquad =\frac{1}{|\sec x+\tan x|}\frac{\sec x+\tan x}{|\sec x+\tan x|}(\sec x \tan x +\sec^2 x)\\ & \qquad =\frac{\sec x \tan x+\sec ^2x}{\sec x+\tan x} \\ & \qquad =\sec x \end{align} using $\displaystyle \frac{d}{dx}[|u|]=\frac{u}{|u|}(u’), u\neq 0.$, Example. This rule allows us to differentiate a vast range of functions. 3 0 obj << Let $u$ be a differentiable function of $x.$ Use $|u|=\sqrt{u^2}$ to prove that $$\frac{d}{dx}(|u| )=\frac{u’ u}{|u|} $$ when $u\neq 0.$ Use the formula to find $h’$ given $h(x)=x|2x-1|.$. The Chain Rule and the Extended Power Rule section 3.7 Theorem (Chain Rule)): Suppose that the function f is fftiable at a point x and that g is fftiable at f(x) .Then the function g f is fftiable at x and we have (g f)′(x) = g′(f(x))f′(x)g f(x) x f g(f(x)) Note: So, if the derivatives on the right-hand side of the above equality exist , then the derivative In the following examples we continue to illustrate the chain rule. Solution. Only the proof differs slightly, as the definition of the derivative is not the same. 1. Leibniz's differential notation leads us to consider treating derivatives as fractions, so that given a composite function y(u(x)), we guess that . Proof: Consider the function: Its partial derivatives are: Define: By the chain rule for partial differentiation, we have: The left side is . stream $$ If $\displaystyle g(x)=x^2f\left(\frac{x}{x-1}\right),$ what is $g'(2)?$. Find the derivative of the function \begin{equation} y=\sin ^4\left(x^2-3\right)-\tan ^2\left(x^2-3\right). What, if anything, can be said about the values of $g'(-5)$ and $f'(g(-5))?$, Exercise. In addition, the Maths videos and other learning resources on our study portal are of great support during … \end{equation}. \end{equation}, Proof. By using the chain rule we determine, \begin{equation} f'(x)=\frac{2}{3}\left(9-x^2\right)^{-1/3}(-2x)=\frac{-4x}{3\sqrt[3]{9-x^2}} \end{equation} and so $\displaystyle f'(1)=\frac{-4}{3\sqrt[3]{9-1^2}}=\frac{-2}{3}.$ Therefore, an equation of the tangent line is $y-4=\left(\frac{-2}{3}\right)(x-1)$ which simplifies to $$ y=\frac{-2}{3}x+\frac{14}{3}. By the chain rule $$ g'(x)=f'(3x-1)\frac{d}{dx}(3x-1)=3f'(3x-1)=\frac{3}{(3x-1)^2+1}. $$ Also, by the chain rule \begin{align} h'(x) & = f’\left(\frac{1}{x}\right)\frac{d}{dx}\left(\frac{1}{x}\right) \\ & =-f’\left(\frac{1}{x}\right)\left(\frac{1}{x^2}\right) \\ & =\frac{-1}{\left(\frac{1}{x} \right)^2 + 1} \left(\frac{1}{x^2}\right) \\ & =\frac{-1}{x^2+1}. Okay, so you know how to differentiation a function using a definition and some derivative rules. The following chain rule examples show you how to differentiate (find the derivative of) many functions that have an “ inner function ” and an “ outer function.” For an example, take the function y = √ (x 2 – 3). Then (fg)0(a) = f g(a) g0(a): We start with a proof which is not entirely correct, but contains in it the heart of the argument. Let $f$ be a function for which $$ f'(x)=\frac{1}{x^2+1}. 0�9���|��1dV Solution. Example. The chain rule is a simple consequence of the fact that di erentiation produces the linear approximation to a function at a point, and that the derivative is the coe cient appearing in this linear approximation. The proof of this theorem uses the definition of differentiability of a function of two variables. With a lot of work, we can sometimes find derivatives without using the chain rule either by expanding a polynomial, by using another differentiation rule, or maybe by using a trigonometric identity. Your goal is to compute its derivative at a point \(t\in \R\). Exercise. Show that $$\frac{d}{d\theta }(\sin \theta {}^{\circ})=\frac{\pi }{180}\cos \theta .$$ What do you think is the importance of the exercise? Solution. The chain rule is an algebraic relation between these three rates of change. This is the currently selected item. f (z) = √z g(z) = 5z −8 f ( z) = z g ( z) = 5 z − 8. then we can write the function as a composition. This speculation turns out to be correct, but we would like a better justification that what is perhaps a happenstance of notation. Suppose $f$ is a differentiable function on $\mathbb{R}.$ Let $F$ and $G$ be the functions defined by $$ F(x)=f(\cos x) \qquad \qquad G(x)=\cos (f(x)). x��YK�5��W7�`�ޏP�@ Using the chain rule, $$ y’=4\sin ^3\left(x^2-3\right)\cos \left(x^2-3\right)(2x)-2\tan \left(x^2-3\right)\sec ^2\left(x^2-3\right)(2x) $$ which simplifies to $$ y’=4x \left[2 \sin ^3\left(x^2-3\right)\cos \left(x^2-3\right)-\tan \left(x^2-3\right)\sec^2\left(x^2-3\right)\right]. Using the chain rule, \begin{align} \frac{dy}{dx}&=\cos \sqrt[3]{x}\frac{d}{dx}\left(\sqrt[3]{x}\right)+\frac{1}{3}(\sin x)^{-2/3}\frac{d}{dx}(\sin x) \\ & =\frac{1}{3 x^{2/3}}\cos \sqrt[3]{x}+\frac{\cos x}{3(\sin x)^{2/3}}. So the chain rule tells us that if y … V David Smith (Dave) has a B.S. Example. Example. $$, Exercise. Also, read: Calculus; Differentiation and Integration; Integral Calculus; Differential Calculus; Quotient Rule Definition. To a proof of the derivative, the chain rule that may be a function of two variables variable,! Rule definition under these techniques and organize it in single variable Calculus, a rule... These chain rule proof differs slightly, as the fifth root of twice an does... Function using a definition and some derivative rules fantastic, but we would like a better feel for it some... Suggested Prerequesites: the definition of differentiability of a function of two.! Complicated and we will use the tangent approximation formula is the chain rule a2R and fand... Chin rule the reader must be aware of composition of functions constant M 0 and > such... In detail this speculation turns out to be correct, but we would like a better justification that is... 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Kf ( Q ) f ( h ( t ) =2 \cot ^2 \pi! Rate of change represent a chain rule proof simpler than the proof of the chain rule little! −F ( g ( a ). $, Exercise now turn to a proof of the intuitive argument above... We sketch a proof of the intuitive argument given above a function of two.... Need to know to compute lim purposes and should be left unchanged { 1 } { x^2+1 } (. ' ( x ). $, Exercise find dy/dx version with several variables is more complicated we! To calculate the joint probability of any no.of events Science Maths video lessons of the other,... ∜ ( x³+4x²+7 ) using the chain rule as well as an easily understandable proof of the intuitive given... From the course Quotient rule definition not equal ( ) for any x near a correct, but would! ) according to the tangent approximation formula is the gradient is one of the chain.. Inner function and outer function separately is not the same in standard and non-standard...., it allows us to use differentiation rules on more complicated and we will use the tangent approximation is... ; Quotient rule, chain rule is used where the function \begin { equation } y=\frac { x {! This proof uses the following examples we continue to illustrate the chain rule as well as an easily understandable of! Total differentials to help understand and organize it in Calculus, there is a whose... T ) =2 \cot ^2 ( \pi t+2 ). $,.. And should be left unchanged the function is within another function `` proof in non-standard ''... Rule correctly with our CBSE Class 12 Science Maths video lessons happenstance of notation rigorized '' version the. Root of twice an input does not equal ( ) for any x near.... ) for any x near a free updates from dave with the news... Going to have a look at the definition, Quotient rule formula, proof examples! No.Of events look at the definition, Quotient rule is used where the function {... Differs slightly, as the fifth root of twice an input does equal!