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Dave4Math » Calculus 1 » The Chain Rule (Examples and Proof). The general power rule states that this derivative is n times the function raised to the (n-1)th power … What, if anything, can be said about the values of $g'(-5)$ and $f'(g(-5))?$, Exercise. Copyright 2010- 2017 MathBootCamps | Privacy Policy, Click to share on Twitter (Opens in new window), Click to share on Facebook (Opens in new window), Click to share on Google+ (Opens in new window), https://www.mathbootcamps.com/derivative-natural-log-lnx/. Therefore, the rule for differentiating a composite function is often called the chain rule. Example: Chain rule for f(x,y) when y is a function of x The heading says it all: we want to know how f(x,y)changeswhenx and y change but there is really only one independent variable, say x,andy is a function of x. However, that is not always the case. ⁡. After having gone through the stuff given above, we hope that the students would have understood, "Chain Rule Examples With Solutions"Apart from the stuff given in "Chain Rule Examples With Solutions", if you need any other stuff in math, please use our google custom search here. It窶冱 just like the ordinary chain rule. The chain rule is similar to the product rule and the quotient rule, but it deals with differentiating compositions of functions. Therefore, the rule for differentiating a composite function is often called the chain rule. 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. 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}. Using the point-slope form of a line, an equation of this tangent line is or . Chain Rule: The General Power Rule The general power rule is a special case of the chain rule. This section gives plenty of examples of the use of the chain rule as well as an easily understandable proof of the chain rule. This is more formally stated as, if the functions f (x) and g (x) are both differentiable and define F (x) = (f o g)(x), then the required derivative of the function F(x) is, This formal approach is defined for a differentiation of function of a function. Find the derivative of the function \begin{equation} h(t)=2 \cot ^2(\pi t+2). The derivative would be the same in either approach; however, the chain rule allows us to find derivatives that would otherwise be very difficult to handle. If x + 3 = u then the outer function becomes f = u 2. Show Solution We’ve already identified the two functions that we needed for the composition, but let’s write them back down anyway and take their derivatives. The same idea will work here. In this example, there is a function $$3x+1$$ that is being taken to the 5th power. The chain rule tells us how to find the derivative of a composite function. Choose your video style (lightboard, screencast, or markerboard), 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? $$f'(x) = \boxed{5(3x+1)^4(3) = 15(3x+1)^4}$$. Examples Problems in Differentiation Using Chain Rule Question 1 : Differentiate y = (1 + cos 2 x) 6 Let f(x)=6x+3 and g(x)=−2x+5. If you're seeing this message, it means we're having trouble loading external resources on our website. 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). Assume that t seconds after his jump, his height above sea level in meters is given by g(t) = 4000 − 4.9t 2. Exercise. Comments are currently disabled. 165-171 and A44-A46, 1999. Example. \end{equation} as desired. Let u = x2 so that y = cosu. 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 outer layer is the square'' and the inner layer is (3 x +1). The only deal is, you will have to pay a penalty. If x + 3 = u then the outer function becomes f = u 2. Chain Rule Help. Let f be a function for which f(2)=-3 and f'(x)=\sqrt{x^2+5}. About this resource. Example. In school, there are some chocolates for 240 adults and 400 children. The multivariable chain rule is more often expressed in terms of the gradient and a vector-valued derivative. Topics. This section gives plenty of examples of the use of the chain rule as well as an easily understandable proof of the chain rule. 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… Normally, if it was just $$\ln(x)$$, you would say the derivative is $$\dfrac{1}{x}$$. Differentiation Using the Chain Rule. When you apply one function to the results of another function, you create a composition of functions. (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 \begin{equation} \frac{d f}{d x}=\frac{df}{du}\frac{du}{dx}. Proof of the chain rule. 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. That material is here.. Want to skip the Summary? Show that \begin{equation} \frac{d}{d x}( \ln |\cos x| )=-\tan x \qquad \text{and}\qquad \frac{d}{d x}(\ln|\sec x+\tan x|)=\sec x. The capital F means the same thing as lower case f, it just encompasses the composition of functions. Are you working to calculate derivatives using the Chain Rule in Calculus? So let’s dive right into it! Then justify your claim. §3.5 and AIII in Calculus with Analytic Geometry, 2nd ed. So, there are two pieces: the $$3x + 1$$ (the inside function) and taking that to the 5th power (the outside function). The chain rule tells us how to find the derivative of a composite function. Table of Contents, The chain rule says that if $$h$$ and $$g$$ are functions and $$f(x) = g(h(x))$$, then. The chain rule has many applications in Chemistry because many equations in Chemistry describe how one physical quantity depends on another, which in turn depends on another. 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$. To prove the chain rule let us go back to basics. Solution. \end{align} as needed. Using the chain rule and the quotient rule, \begin{equation} \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} \end{equation} which simplifies to \begin{equation} \frac{dy}{dx}=\frac{4-x^4}{\left(4+x^4\right)^{3/2}} \end{equation} as desired. Tree diagrams are useful for deriving formulas for the chain rule for functions of more than one variable, where each independent variable also depends on other variables. Okay, so you know how to differentiation a function using a definition and some derivative rules. Example. And, in the nextexample, the only way to obtain the answer is to use the chain rule. 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{y+\sqrt{2y-9}}\right)^8$, Exercise. Find the derivative of $$f(x)=\ln(x^2-1)$$. Click HERE for a real-world example of the chain rule. This section presents examples of the chain rule in kinematics and simple harmonic motion. Under certain conditions, such as differentiability, the result is fantastic, but you should practice using it. Apostol, T. M. "The Chain Rule for Differentiating Composite Functions" and "Applications of the Chain Rule. In the following examples we continue to illustrate the chain rule. The Chain Rule is a formula for computing the derivative of the composition of two or more functions. Also, read Differentiation method here at BYJU’S. Some of the types of chain rule problems that are asked in the exam. Differentiation Using the Chain Rule SOLUTION 1 : Differentiate. $$If \displaystyle g(x)=x^2f\left(\frac{x}{x-1}\right), what is g'(2)?. Derivative Rules. REFERENCES: Anton, H. "The Chain Rule" and "Proof of the Chain Rule." Composite functions come in all kinds of forms so you must learn to look at functions differently. The Chain Rule is a formula for computing the derivative of the composition of two or more functions. It follows immediately that du dx = 2x dy du = −sinu The chain rule says dy dx = dy du × du dx and so dy dx = −sinu× 2x = −2xsinx2 Example Suppose we want to diﬀerentiate y = cos2 x = (cosx)2. The single-variable chain rule. The Formula for the Chain Rule. From there, it is just about going along with the formula. For instance, if f and g are functions, then the chain rule expresses the derivative of their composition. There are rules we can follow to find many derivatives.. For example: The slope of a constant value (like 3) is always 0; The slope of a line like 2x is 2, or 3x is 3 etc; and so on. Powered by Create your own unique website with customizable templates. This indicates that the function f(x), the inner function, must be calculated before the value of g(x), the outer function, … Exercise. Example. Using the chain rule, \begin{equation} \frac{d}{d x}f'[f(x)] =f” [ f(x)] f'(x) \end{equation} which is the second derivative evaluated at the function multiplied by the first derivative; while, \begin{equation} \frac{d}{d x}f [f'(x)]=f'[f'(x)]f”(x) \end{equation} is the first derivative evaluated at the first derivative multiplied by the second derivative. Find the derivative of the function \begin{equation} y=\sin ^4\left(x^2-3\right)-\tan ^2\left(x^2-3\right). Theorem. Comments. Question 1 . Solution. Find an equation of the tangent line to the graph of the function f(x)=\left(9-x^2\right)^{2/3} at the point (1,4)., Solution. and M.S. The main function $$f(x)$$ is formed by plugging $$h(x)$$ into the function $$g$$. Example. Find the derivative of the function \begin{equation} y=\sin \sqrt{x}+\sqrt{\sin x} \end{equation}, Solution.$$ as desired. In the limit as Δt → 0 we get the chain rule. By the chain rule $$g'(x)=f'(3x-1)\frac{d}{dx}(3x-1)=3f'(3x-1)=\frac{3}{(3x-1)^2+1}. Suppose that u=g(x) is differentiable at x=-5, y=f(u) is differentiable at u=g(-5), and (f\circ g)'(-5) is negative. This section gives plenty of examples of the use of the chain rule as well as an easily understandable proof of the chain rule. The proof of the chain rule is a bit tricky - I left it for the appendix. Chain Rule Examples (both methods) doc, 170 KB. Thus, the slope of the line tangent to the graph of h at x=0 is . Suppose f is a differentiable function on \mathbb{R}. Let F and G be the functions defined by$$ F(x)=f(\cos x) \qquad \qquad G(x)=\cos (f(x)). A few are somewhat challenging. Find the derivative of the function \begin{equation} g(x)=\left(\frac{3x^2-2}{2x+3}\right)^3. When trying to decide if the chain rule makes sense for a particular problem, pay attention to functions that have something more complicated than the usual $$x$$. \end{equation}. Example Suppose we want to diﬀerentiate y = cosx2. This 105. is captured by the third of the … Chain rule Statement Examples Table of Contents JJ II J I Page4of8 Back Print Version Home Page d dx [ f(g x))] = 0 ( gx)) 0(x) # # # d dx sin5 x = 5(sinx)4 cosx 21.2.4 Example Find the derivative d dx h 5x2 4x+3 i. \end{align} as desired. Let $f$ be a function for which  f'(x)=\frac{1}{x^2+1}. 7W ) r ( w ) = ( 6 x 2 + 7 x ) = csc 7w. Equation of this tangent line is $( 1,1 ).$ Exercise. Can actually use the product rule and the quotient rule, that the derivative of h.. 1,1 ). $, Exercise variable involves the partial derivatives with respect to chain.! 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