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Chapter 3: Problem 2

How do you find the derivative of the quotient of two functions that are differentiable at a point?

Short Answer

Expert verified
Question: Using the Quotient Rule, find the derivative of the function h(x) = (3x^2 + x) / (x^2 - 1). Answer: h'(x) = (-2x^4 + x^2 - 6x - 1) / (x^4 - 2x^2 + 1)

Step by step solution

01

Recall the Quotient Rule formula

The Quotient Rule states that if we have two functions, f(x) and g(x), and we want to find the derivative of their quotient h(x) = f(x) / g(x), then the derivative h'(x) can be found using the formula: h'(x) = (g(x) * f'(x) - f(x) * g'(x)) / (g(x))^2
02

Identify the two functions

Identify the two functions f(x) and g(x) which form the quotient. For instance, if given the problem h(x) = (3x^2 + x) / (x^2 - 1), then f(x) = 3x^2 + x and g(x) = x^2 - 1.
03

Find their derivatives

Use the rules of differentiation to find the derivatives of the functions f(x) and g(x) separately. Continuing with the example given in Step 2: f'(x) = d/dx (3x^2 + x) = 6x + 1 g'(x) = d/dx (x^2 - 1) = 2x
04

Apply the Quotient Rule formula

Substitute the functions f(x), g(x) and their derivatives f'(x), g'(x) into the Quotient Rule formula to find h'(x): h'(x) = ((x^2 - 1) * (6x + 1) - (3x^2 + x) * (2x)) / (x^2 - 1)^2
05

Simplify the expression

Simplify the resulting expression to find the derivative in its simplest form. In the given example: h'(x) = (6x^3 + x^2 - 6x - 1 - 6x^3 - 2x^4) / (x^4 - 2x^2 + 1) = (-2x^4 + x^2 - 6x - 1) / (x^4 - 2x^2 + 1) Now we have found the derivative of the quotient of the two functions, h'(x) = (-2x^4 + x^2 - 6x - 1) / (x^4 - 2x^2 + 1).

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Most popular questions from this chapter

A trough in the shape of a half cylinder has length \(5 \mathrm{m}\) and radius \(1 \mathrm{m}\). The trough is full of water when a valve is opened, and water flows out of the bottom of the trough at a rate of \(1.5 \mathrm{m}^{3} / \mathrm{hr}\) (see figure). (Hint: The area of a sector of a circle of a radius \(r\) subtended by an angle \(\theta\) is \(r^{2} \theta / 2 .\) ) a. How fast is the water level changing when the water level is \(0.5 \mathrm{m}\) from the bottom of the trough? b. What is the rate of change of the surface area of the water when the water is \(0.5 \mathrm{m}\) deep?

a. Determine an equation of the tangent line and normal line at the given point \(\left(x_{0}, y_{0}\right)\) on the following curves. b. Graph the tangent and normal lines on the given graph. \(\left(x^{2}+y^{2}\right)^{2}=\frac{25}{3}\left(x^{2}-y^{2}\right)\) \(\left(x_{0}, y_{0}\right)=(2,-1)\) (lemniscate of Bernoulli)

Identifying functions from an equation. The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for \(y\) to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots\) c. Use the functions found in part (b) to graph the given equation. $$x^{4}=2\left(x^{2}-y^{2}\right) \text { (eight curve) }$$

Recall that \(f\) is even if \(f(-x)=f(x),\) for all \(x\) in the domain of \(f,\) and \(f\) is odd if \(f(-x)=-f(x),\) for all \(x\) in the domain of \(f\). a. If \(f\) is a differentiable, even function on its domain, determine whether \(f^{\prime}\) is even, odd, or neither. b. If \(f\) is a differentiable, odd function on its domain, determine whether \(f^{\prime}\) is even, odd, or neither.

Find the following higher-order derivatives. $$\frac{d^{n}}{d x^{n}}\left(2^{x}\right)$$
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