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Chapter 2: Problem 116

Using Trigonometric Functions (a) Find the derivative of the function \(g(x)=\sin ^{2} x+\cos ^{2} x\) in two ways. (b) For \(f(x)=\sec ^{2} x\) and \(g(x)=\tan ^{2} x,\) show that \(f^{\prime}(x)=g^{\prime}(x)\)

Short Answer

Expert verified
The derivative of \(g(x) = \sin^2 x + \cos^2 x\) is 0. The derivatives of \(f(x) = \sec^2 x\) and \(g(x) = \tan^2 x\) are both equal to \(2 \sec^2x \tan x\).

Step by step solution

01

First derivative for part (a)

For the function \(g(x) = \sin^2 x + \cos^2 x\), note that this sum \(sin^2x + cos^2x\) is a well-known identity that equals to 1. Therefore, \(g(x) = 1\), and the derivative of a constant is 0. Hence, \(g'(x) = 0\).
02

Second derivative for part (a)

For a deeper understanding, we can differentiate the given function using the chain rule too,\[\frac{d}{dx}(\sin^2x) = 2\sin x \cos x\]\[\frac{d}{dx}(\cos^2x) = -2\sin x \cos x\]So \(g'(x) = 2\sin x \cos x - 2\sin x \cos x = 0\]
03

Derivative for \(f(x)\) in part (b)

For the function \(f(x) = \sec^2 x\), the derivative is calculated as \(f'(x) = 2 \sec{x} \cdot \sec{x} \tan{x}\)
04

Derivative for \(g(x)\) in part (b)

For the function \(g(x) = \tan^2 x\), the derivative is calculated as \(g'(x) = 2 \tan{x} \cdot \sec^2{x}\)
05

Verify the equality of derivatives for part (b)

From step 3 and 4, it can be seen that both \(f'(x)\) and \(g'(x)\) simplify to \(2 \sec^2{x} \tan{x}\) therefore we confirm that \(f'(x)=g'(x)\)

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Most popular questions from this chapter

Determining Differentiability In Exercises \(85-88\) , find the derivatives from the left and from the right at \(x=1\) (if they exist). Is the function differentiable at \(x=1 ?\) $$ f(x)=\sqrt{1-x^{2}} $$

Moving Point In Exercises \(5-8,\) a point is moving along the graph of the given function at the rate \(d x / d t .\) Find \(d y / d t\) for the given values of \(x .\) $$ \begin{array}{l}{y=\tan x ; \frac{d x}{d t}=3 \text { feet per second }} \\\ {\begin{array}{llll}{\text { (a) } x=-\frac{\pi}{3}} & {\text { (b) } x=-\frac{\pi}{4}} & {\text { (c) } x=0}\end{array}}\end{array} $$

A trough is 12 feet long and 3 feet across the top (see figure). Its ends are isosceles triangles with altitudes of 3 feet. (a) Water is being pumped into the trough at 2 cubic feet per minute. How fast is the water level rising when the depth \(h\) is 1 foot? (b) The water is rising at a rate of \(\frac{3}{8}\) inch per minute when \(h=2 .\) Determine the rate at which water is being pumped into the trough.

Modeling Data The table shows the health care expenditures \(h\) (in billions of dollars) in the United States and the population \(p\) (in millions) of the United States for the years 2004 through 2009 . The year is represented by \(t,\) with \(t=4\) corresponding to 2004 . (Source: U.S. Centers for Medicare \& Medicaid Services and U.S. Census Bureau) $$ \begin{array}{|c|c|c|c|c|c|}\hline \text { Year, } & {4} & {5} & {6} & {7} & {8} & {9} \\ \hline h & {1773} & {1890} & {2017} & {2135} & {2234} & {2330} \\\ \hline p & {293} & {296} & {299} & {302} & {305} & {307} \\\ \hline\end{array} $$ (a) Use a graphing utility to find linear models for the health care expenditures \(h(t)\) and the population \(p(t) .\) (b) Use a graphing utility to graph each model found in part (a). (c) Find \(A=h(t) / p(t),\) then graph \(A\) using a graphing utility. What does this function represent? (d) Find and interpret \(A^{\prime}(t)\) in the context of these data.

Rate of Change Determine whether there exist any values of \(x\) in the interval \([0,2 \pi)\) such that the rate of change of \(f(x)=\sec x\) and the rate of change of \(g(x)=\csc x\) are equal.
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