The proof of this theorem uses the definition of differentiability of a function of two variables. . This is the currently selected item. This proof uses the following fact: Assume , and . $$If g(x)=f(3x-1), what is g'(x)? Also, if  h(x)=f\left(\frac{1}{x}\right), what is h'(x)?. Suppose that f is differentiable at the point $$\displaystyle P(x_0,y_0),$$ where $$\displaystyle x_0=g(t_0)$$ and … Also related to the tangent approximation formula is the gradient of a function. Exercise. Then using the definition of the derivative, we can write u'(x) as: u′(x)=Limx→0u(x+h)–u(x)h{u'(x) = Lim_{x\rightarrow{0}}\frac{u(x + h) – u(x)}{h}} u′(x)=Limx→0​hu(x+h)–… Find the derivative of the following functions.(1) \quad \displaystyle r=-(\sec \theta +\tan \theta )^{-1}$$(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. \end{align} as desired. Dave will teach you what you need to know. In fact, the chain rule says that the first rate of change is the product of the other two. 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 The standard proof of the multi-dimensional chain rule can be thought of in this way. Using the chain rule and the quotient rule, we determine, $$\frac{dg}{dx} =3\left(\frac{3x^2-2}{2x+3} \right)^2\left(\frac{(2x+3)6x-\left(3x^2-2\right)2}{(2x+3)^2}\right)$$ which simplifies to $$\frac{dg}{dx}=\frac{6 \left(2-3 x^2\right)^2 \left(2+9 x+3 x^2\right)}{(3+2 x)^4}$$ as desired. We will need: Lemma 12.4. The following is a proof of the multi-variable Chain Rule. 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)). 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. Solution. $$, Exercise. Lecture 3: Chain Rules and Inequalities Last lecture: entropy and mutual information This time { Chain rules { Jensen’s inequality { Log-sum inequality { Concavity of entropy { Convex/concavity of mutual information Dr. Yao Xie, ECE587, Information Theory, Duke University To prove: wherever the right side makes sense. Find the derivative of the function y=\sin ^4\left(x^2-3\right)-\tan ^2\left(x^2-3\right). With a lot of work, we can sometimes find derivatives without using the chain rule either by expanding a polynomial, by using another differentiation rule, or maybe by using a trigonometric identity. In Calculus, a Quotient rule is similar to the product rule. Purported Proof of the Chain Rule: Recall that dy du = f0(u) = lim ∆u→0 f(u+∆u)−f(u) ∆u and let u = g(x) and ∆u = ∆g = g(x+∆x)−g(x). By the chain rule$$ g'(x)=f'(3x-1)\frac{d}{dx}(3x-1)=3f'(3x-1)=\frac{3}{(3x-1)^2+1}. Find the derivative of the function y=\frac{x}{\sqrt{x^4+4}}. h→0. Receive free updates from Dave with the latest news! We wish to show $\frac{d f}{d x}=\frac{df}{du}\frac{du}{dx}$ and will do so by using the definition of the derivative for the function $f$ with respect to $x,$ namely, \frac{df}{dx}=\lim_{\Delta x\to 0}\frac{f[u(x+\Delta x)]-f[u(x)]}{\Delta x} To better work with this limit let’s define an auxiliary function: $$g(t)= \begin{cases} \displaystyle \frac{f[u(x)+t]-f[u(x)]}{t}-\frac{df}{du} & \text{ if } t\neq 0 \\ 0 & \text{ if } t=0 \end{cases}$$ Let $\Delta u=u(x+\Delta x)-u(x),$ then three properties of the function $g$ are. To begin with, let us introduce a variable u = g(x) to simplify the looks of our steps. However, we can get a better feel for it using some intuition and a couple of examples. 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 Dave4Math » Calculus 1 » The Chain Rule (Examples and Proof). Thus, $$\frac{dv}{d s}=\frac{-12t+8}{-6t^2+8t+1}. Example. Theorem. Evaluating Limits Analytically (Using Limit Theorems) [Video], Intuitive Introduction to Limits (The Behavior of a Function) [Video], Related Rates (Applying Implicit Differentiation), Numerical Integration (Trapezoidal and Simpson’s), Integral Definition (The Definite Integral), Indefinite Integrals (What is an antiderivative? Chain Rule If f(x) and g(x) are both differentiable functions and we define F(x) = (f ∘ g)(x) then the derivative of F (x) is F ′ (x) = f ′ (g(x)) g ′ (x). 5 Stars: 5: … Solution. Example. It is especially transparent using o() notation, where once again f(x) = o(g(x)) means that lim x!0 f(x) g(x) = 0:$$, Solution. Show that \frac{d}{d x}( \ln |\cos x| )=-\tan x \qquad \text{and}\qquad \frac{d}{d x}(\ln|\sec x+\tan x|)=\sec x. Let's … The chain rule gives another method to find the derivative of a function whose input is another function. Customer reviews (1) 5,0 of 5 stars. Chain rule is a formula which is the same in standard and non-standard analysis. Next lesson. The outer function is √ (x). In other words, we want to compute lim. Determine the point(s) at which the graph of f(x)=\frac{x}{\sqrt{2x-1}} has a horizontal tangent. A new subsection, called "Proof in non-standard analysis", of the section "Proofs" could be added. The chain rule is an algebraic relation between these three rates of change. First proof. (Chain Rule) Suppose $f$ is a differentiable function of $u$ which is a differentiable function of $x.$ Then $f(u(x))$ is a differentiable function of $x$ and \frac{d f}{d x}=\frac{df}{du}\frac{du}{dx}. and M.S. 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|.. \end{align} as desired. /Length 1995 Copyright © 2020 Dave4Math LLC. So the chain rule tells us that if y … Solution. Differentiate the functions given by the following equations (1) \quad y=\cos^2\left(\frac{1-\sqrt{x}}{1+\sqrt{x}}\right)(2) \quad y=\sqrt{1+\tan \left(x+\frac{1}{x}\right)} (3) \quad n=\left(y+\sqrt[3]{y+\sqrt{2y-9}}\right)^8, Exercise. ... » Session 36: Proof » Session 37: …. If $g(t)=[f(\sin t)]^2,$ where $f$ is a differentiable function, find $g'(t).$, Exercise. Let AˆRn be an open subset and let f: A! Using the differentiation rule $\frac{d}{dx}[\ln u]=\frac{u’}{u};$ we have, $$\frac{d}{d x}( \ln |\cos x| ) =\frac{1}{\cos x}\frac{d}{dx}(\cos x) =\frac{\sin x}{\cos x} =\tan x$$ 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. Suppose that the functions $f$, $g$, and their derivatives with respect to $x$ have the following values at $x=0$ and $x=1.$ $$\begin{array}{c|cccc} x & f(x) & g(x) & f'(x) & g'(x) \\ \hline 0 & 1 & 1 & 5 & 1/3 \\ 1 & 3 & -4 & -1/3 & -8/3 \end{array}$$ 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$. \end{align} as needed. In addition, the Maths videos and other learning resources on our study portal are of great support during … Proof of the Chain Rule • Given two functions f and g where g is diﬀerentiable at the point x and f is diﬀerentiable at the point g(x) = y, we want to compute the derivative of the composite function f(g(x)) at the point x. Then justify your claim. Proof. Assuming that the following derivatives exists, find \frac{d}{d x}f’ [f(x)] \qquad \text{and}\qquad \frac{d}{d x}f [f'(x)]. 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}. Included Quiz Questions. 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. If \displaystyle g(x)=x^2f\left(\frac{x}{x-1}\right), what is g'(2)?. Find an equation of the tangent line to the graph of the function f(x)=\left(9-x^2\right)^{2/3} at the point (1,4)., Solution. David is the founder and CEO of Dave4Math. "7�� 7�n��6��x�;�g�P��0ݣr!9~��g�.X�xV����;�T>�w������tc�y�q���%[c�lC�ŵ�{HO;���v�~�7�mr � lBD��. f(g(x+h))−f(g(x)) h . The version with several variables is more complicated and we will use the tangent approximation and total differentials to help understand and organize it. In differential calculus, the chain rule is a way of finding the derivative of a function. What does this rate of change represent? Example. . Using the chain rule and the quotient rule, $$\frac{dy}{dx}=\frac{\sqrt{x^4+4}(1)-x\frac{d}{dx}\left(\sqrt{x^4+4}\right)}{\left(\sqrt{x^4+4}\right)^2}=\frac{\sqrt{x^4+4}(1)-x\left(\frac{2 x^3}{\sqrt{4+x^4}}\right)}{\left(\sqrt{x^4+4}\right)^2}$$ which simplifies to $$\frac{dy}{dx}=\frac{4-x^4}{\left(4+x^4\right)^{3/2}}$$ as desired. . Proof. It is used where the function is within another function. Understand how to differentiate composite functions by using the Chain Rule correctly with our CBSE Class 12 Science Maths video lessons. A Quotient Rule is stated as the ratio of the quantity of the denominator times the derivative of the numerator function … 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 $s(t)=-2t^3+4t^2+t-3.$. The lecture Chain Rule Proof by Batool Akmal is from the course Quotient Rule, Chain Rule and Product Rule. Example. The right side becomes: This simplifies to: Plug back the expressions and get: 1. $$. Find the derivative of the function $$y=\sin \sqrt[3]{x}+\sqrt[3]{\sin x}$$, Solution. Show that$$\frac{d}{d\theta }(\sin \theta {}^{\circ})=\frac{\pi }{180}\cos \theta .$$What do you think is the importance of the exercise? Determine if the following statement is true or false.$$ as desired. Batool Akmal. Using the chain rule and the product rule we determine, $$g'(x)=2x f\left(\frac{x}{x-1}\right)+x^2f’\left(\frac{x}{x-1}\right)\frac{d}{dx}\left(\frac{x}{x-1}\right)$$ = 2x f\left(\frac{x}{x-1}\right)+x^2f’\left(\frac{x}{x-1}\right)\left(\frac{-1}{(x-1)^2}\right). The chain rule is used to differentiate composite functions. Solution. Then the previous expression is equal to the product of two factors: Your goal is to compute its derivative at a point $$t\in \R$$. f (z) = √z g(z) = 5z −8 f ( z) = z g ( z) = 5 z − 8. then we can write the function as a composition. Proof: Consider the function: Its partial derivatives are: Define: By the chain rule for partial differentiation, we have: The left side is . 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}}. 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. By using the chain rule we determine, f'(x)=\frac{2}{3}\left(9-x^2\right)^{-1/3}(-2x)=\frac{-4x}{3\sqrt[3]{9-x^2}} 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}. Only the proof differs slightly, as the definition of the derivative is not the same. Find the derivative of the function g(x)=\left(\frac{3x^2-2}{2x+3}\right)^3. Here is the chain rule again, still in the prime notation of Lagrange. And with that, we’ll close our little discussion on the theory of Chain Rule as of now. The single-variable chain rule. Also, read: Calculus; Differentiation and Integration; Integral Calculus; Differential Calculus; Quotient Rule Definition. V ), 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). in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. Proving the chain rule. If y = (1 + x²)³ , find dy/dx . Leibniz's differential notation leads us to consider treating derivatives as fractions, so that given a composite function y(u(x)), we guess that . Given: Functions and . What, if anything, can be said about the values of g'(-5) and f'(g(-5))?, Exercise. Derivative rules review. Then is differentiable at if and only if there exists an by matrix such that the "error" function has the property that approaches as approaches . Solution. Example. %PDF-1.4 This field is for validation purposes and should be left unchanged. 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. We now turn to a proof of the chain rule. Assume for the moment that () does not equal () for any x near a. Using the chain rule and the formula \displaystyle \frac{d}{dx}(\cot u)=-u’\csc ^2u, \begin{align} \frac{dh}{dt} & =4\cot (\pi t+2)\frac{d}{dx}[\cot (\pi t+2)] \\ & =-4\pi \cot (\pi t+2)\csc ^2(\pi t+2). Therefore, 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. \end{align}, Example. Let’s see this for the single variable case rst. 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). As fis di erentiable at P, there is a constant >0 such that if k! What is the gradient of y = F(H(x)) according to the chain rule? 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: The proof of the chain rule is a bit tricky - I left it for the appendix. Practice: Chain rule capstone. Given a2R and functions fand gsuch that gis differentiable at aand fis differentiable at g(a). 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 By the chain rule g'(x)=f'(3x-1)\frac{d}{dx}(3x-1)=3f'(3x-1)=\frac{3}{(3x-1)^2+1}. Solution. Let $f$ be a function for which $$f'(x)=\frac{1}{x^2+1}. �Vq ���N�k?H���Z��^y�l6PpYk4ږ�����=_^�>�F�Jh����n� �碲O�_�?�W�Z��j"�793^�_=�����W��������b>���{� =����aޚ(�7.\��� l�����毉t�9ɕ�n"�� ͬ���ny�m��M+��eIǬѭ���n����t9+���l�����]��v���hΌ��Ji6I�Y)H\���f$$ Now we can rewrite $\displaystyle \frac{df}{dx}$ as follows: \begin{align} \frac{df}{dx} & = \lim_{\Delta x\to 0}\frac{f[u(x+\Delta x)]-f[u(x)]}{\Delta x} \\ & =\lim_{\Delta x\to 0}\frac{f[u(x)+\Delta u]-f[u(x)]}{\Delta x} \\ & =\lim_{\Delta x\to 0} \frac{\left(g(\Delta u)+\frac{df}{du}\right)\Delta u}{\Delta x} \\ & =\lim_{\Delta x\to 0}\left(g(\Delta u)+\frac{df}{du}\right)\lim_{\Delta x\to 0}\frac{\Delta u}{\Delta x} \\ & =\left[\lim_{\Delta x\to 0}g(\Delta u)+\lim_{\Delta x\to 0}\frac{df}{du}\right]\text{ }\lim_{\Delta x\to 0}\frac{\Delta u}{\Delta x} \\ & =\left[g\left( \lim_{\Delta x\to 0}\Delta u \right)+\lim_{\Delta x\to 0}\frac{df}{du}\right]\text{ }\lim_{\Delta x\to 0}\frac{\Delta u}{\Delta x} \\ & =\left[g(0)+\lim_{\Delta x\to 0}\frac{df}{du}\right]\text{ }\lim_{\Delta x\to 0}\frac{\Delta u}{\Delta x} \\ & =\left[0+\lim_{\Delta x\to 0}\frac{df}{du}\right]\text{ }\lim_{\Delta x\to 0}\frac{\Delta u}{\Delta x} \\ & =\frac{df}{du}\frac{du}{dx}. It's a "rigorized" version of the intuitive argument given above. Example. /Filter /FlateDecode This speculation turns out to be correct, but we would like a better justification that what is perhaps a happenstance of notation. 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}$$(6) \quad \displaystyle y=e^x \sqrt{2x-x^2}, Exercise. f [ g ( x)] – f [ g ( c)] x – c = Q [ g ( x)] g ( x) − g ( c) x − c. for all x in a punctured neighborhood of c. In which case, the proof of Chain Rule can be finalized in a few steps through the use of limit laws. On the other hand, simple basic functions such as the fifth root of twice an input does not fall under these techniques. Proof of the Chain Rule •Suppose u = g(x) is differentiable at a and y = f(u) is differentiable at b = g(a). Let f be a function for which f(2)=-3 and$$ f'(x)=\sqrt{x^2+5}. In this article, you are going to have a look at the definition, quotient rule formula, proof and examples in detail. Chain rule capstone. Define $$\phi = f\circ \mathbf g$$. PQk: Proof. stream By the chain rule, a(t)=\frac{dv}{dt}=\frac{dv}{d s}\frac{ds}{dt}=v(t)\frac{dv}{ds} In the case where$s(t)=-2t^3+4t^2+t-3; $we determine, \frac{ds}{dt} = v(t) = -6t^2+8t+1 \qquad \text{and } \qquad a(t)=-12t+8. In the following examples we continue to illustrate the chain rule. Find the derivative of the function h(t)=2 \cot ^2(\pi t+2). Suppose that$u=g(x)$is differentiable at$x=-5,y=f(u)$is differentiable at$u=g(-5),$and$(f\circ g)'(-5)is negative. PQk< , then kf(Q) f(P) Df(P)! Example. Worked example: Derivative of ∜(x³+4x²+7) using the chain rule. The inner function is the one inside the parentheses: x 2 -3. 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$f'(x)=\frac{-1+x}{(-1+2 x)^{3/2}}. Example. R(z) = (f ∘g)(z) = f (g(z)) = √5z−8 R ( z) = ( f ∘ g) ( z) = f ( g ( z)) = 5 z − 8. and it turns out that it’s actually fairly simple to differentiate a function composition using the Chain Rule. Example. When will these derivatives be the same? x��YK�5��W7�`�ޏP�@ The chain rule can be used iteratively to calculate the joint probability of any no.of events. Theorem 1 (Chain Rule). Under certain conditions, such as differentiability, the result is fantastic, but you should practice using it. Up Next. The gradient is one of the key concepts in multivariable calculus. 3 0 obj << If fis di erentiable at P, then there is a constant M 0 and >0 such that if k! (a) Find the tangent to the curve$y=2 \tan (\pi x/4)$at$x=1.$(b) What is the smallest value the slope of the curve can ever have on the interval$-2 0 such that if k multivariable.., still in the prime notation of Lagrange ( h ( t ) \cot! Of examples variable Calculus, there is a multivariable chain rule a better feel for using.