# chain rule examples

This makes it look very analogous to the single-variable chain rule. Solution. \end{equation}. Composite functions come in all kinds of forms so you must learn to look at functions differently. Raj and Isaiah both leave their respective houses at 7 a.m. for their daily run. (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}. \end{equation} Thus, \begin{equation} \frac{dv}{d s}=\frac{-12t+8}{-6t^2+8t+1}. Find the derivative of the function \begin{equation} g(x)=\left(\frac{3x^2-2}{2x+3}\right)^3. 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. In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x) . Exercise. 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. 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. y = 3√1 −8z y = 1 − 8 z 3 Solution. 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. 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. Question 1 . Dave will help you with what you need to know, Calculus (Start Here) – Enter the World of Calculus, Mathematical Proofs (Using Various Methods), Chinese Remainder Theorem (The Definitive Guide), Math Solutions: Step-by-Step Solutions to Your Problems, Math Videos: Custom Made Videos For Your Problems, LaTeX Typesetting: Trusted, Fast, and Accurate, LaTeX Graphics: Custom Graphics Using TikZ and PGFPlots. Now you can simplify to get the final answer: If you need to review taking the derivative of ln(x), see this lesson: https://www.mathbootcamps.com/derivative-natural-log-lnx/. Composite functions come in all kinds of forms so you must learn to look at functions differently. Thus, the slope of the line tangent to the graph of h at x=0 is . 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. Let f(x)=6x+3 and g(x)=−2x+5. Solution. But it is absolutely indispensable in general and later, and already is very helpful in dealing with polynomials. This is called a tree diagram for the chain rule for functions of one variable and it provides a way to remember the formula (Figure $$\PageIndex{1}$$). Click HERE for a real-world example of the chain rule. For example, if a composite function f (x) is defined as Example. Using the chain rule, if you want to find the derivative of the main function $$f(x)$$, you can do this by taking the derivative of the outside function $$g$$ and then multiplying it by the derivative of the inside function $$h$$. The arguments of the functions are linked (chained) so that the value of an internal function is the argument for the following external function. Oct 5, 2015 - Explore Rod Cook's board "Chain Rule" on Pinterest. Suppose we pick an urn at random and then select a … Created: Dec 4, 2011. Copyright © 2021 Dave4Math, LLC. (The outer layer is the square'' and the inner layer is (3 x +1). Differentiation Using the Chain Rule. Chain Rule Help. So, cover it up and take the derivative anyway. Example 1 Use the Chain Rule to differentiate $$R\left( z \right) = \sqrt {5z - 8}$$. Here we are going to see some example problems in differentiation using chain rule. For example, in (11.2), the derivatives du/dt and dv/dt are evaluated at some time t0. What, if anything, can be said about the values of $g'(-5)$ and $f'(g(-5))?$, Exercise. 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 $h(x)=x|2x-1|.$. ⁡. Chain Rule Examples. Partial Derivatives. Brush up on your knowledge of composite functions, and learn how to apply the chain rule correctly. §3.5 and AIII in Calculus with Analytic Geometry, 2nd ed. ), L ‘Hopital’s Rule and Indeterminate Forms, Parametric Equations and Calculus (Finding Tangent Lines), Linearization and Differentials (by Example). The Formula for the Chain Rule. From there, it is just about going along with the formula. Therefore, the rule for differentiating a composite function is often called the chain rule. Raj runs north at 9 km/h, while Isaiah runs west at 7 km/h. Using the chain rule, \begin{align} \frac{dy}{dx}&=\cos \sqrt{x}\frac{d}{dx}\left(\sqrt{x}\right)+\frac{1}{3}(\sin x)^{-2/3}\frac{d}{dx}(\sin x) \\ & =\frac{1}{3 x^{2/3}}\cos \sqrt{x}+\frac{\cos x}{3(\sin x)^{2/3}}. So let’s dive right into it! \end{align} as needed. Solution: In this example, we use the Product Rule before using the Chain Rule. So, cover up that $$3x + 1$$, and pretend it is an $$x$$ for a minute. (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