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Chapter 3: Problem 27

Find \(d y / d x\) for the following functions. $$y=\cos ^{2} x$$

Short Answer

Expert verified
Question: Find the derivative of the function $$y = \cos^2 x$$ with respect to $$x$$. Answer: The derivative of the function $$y = \cos^2 x$$ with respect to $$x$$ is $$\frac{dy}{dx} = -2\cos x \sin x$$.

Step by step solution

01

Identify the outer and inner functions

First, we need to identify the outer and inner functions to apply the chain rule. In this case, the outer function is $$u^2$$, and the inner function is $$u = \cos x$$. So, $$y = u^2$$, where $$u = \cos x$$.
02

Differentiate outer and inner functions

Now, we have to find the derivatives of the outer and inner functions concerning their variables. So, we will find $$\frac{du}{dx}$$ and $$\frac{dy}{du}$$. For the inner function, $$u = \cos x$$, its derivative is $$\frac{du}{dx} = -\sin x$$. For the outer function, $$y = u^2$$, its derivative is $$\frac{dy}{du} = 2u$$.
03

Apply the chain rule

Now, we need to apply the chain rule to find the derivative of the function $$y = \cos^2 x$$. The chain rule states that: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$ Substitute the values we found in Step 2: $$\frac{dy}{dx} = (2u) \cdot (-\sin x)$$
04

Replace $$u$$ with the inner function

To finish finding the derivative, replace $$u$$ with the inner function $$\cos x$$: $$\frac{dy}{dx} = 2(\cos x) \cdot (-\sin x)$$
05

Simplify the expression

Now, simplify the expression: $$\frac{dy}{dx} = -2\cos x \sin x$$ That's the final answer, the derivative of the function $$y = \cos^2 x$$ is: $$\frac{dy}{dx} = -2\cos x \sin x$$

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Most popular questions from this chapter

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. \(x+y^{3}-x y=1\) (Hint: Rewrite as \(y^{3}-1=x y-x\) and then factor both sides.)

Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of \(x,\) and find the derivative of the inverse function. $$f(x)=x^{2}-4, \text { for } x>0$$

Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}(x f(x))\right|_{x=3}$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The derivative \(\frac{d}{d x}\left(e^{5}\right)\) equals \(5 \cdot e^{4}\) b. The Quotient Rule must be used to evaluate \(\frac{d}{d x}\left(\frac{x^{2}+3 x+2}{x}\right)\) c. \(\frac{d}{d x}\left(\frac{1}{x^{5}}\right)=\frac{1}{5 x^{4}}\) d. \(\frac{d^{n}}{d x^{n}}\left(e^{3 x}\right)=3^{n} \cdot e^{3 x},\) for any integer \(n \geq 1\)

The Witch of Agnesi The graph of \(y=\frac{a^{3}}{x^{2}+a^{2}},\) where \(a\) is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi). a. Let \(a=3\) and find an equation of the line tangent to \(y=\frac{27}{x^{2}+9}\) at \(x=2\) b. Plot the function and the tangent line found in part (a).
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