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Chapter 3: Problem 42

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

Short Answer

Expert verified
Answer: The second derivative of the function \(y = \cos x\) is \(y^{\prime\prime} = -\cos x\).

Step by step solution

01

Calculate the first derivative, \(y^{\prime}\)

To find the first derivative, we will apply the differentiation rule for trigonometric functions. In this case, the derivative of \(\cos x\) will be: $$y^{\prime} = -\sin x$$
02

Calculate the second derivative, \(y^{\prime\prime}\)

Now that we have the first derivative, we can differentiate it once more to get the second derivative. The derivative of \(-\sin x\) will be: $$y^{\prime\prime} = -\cos x$$ So, the second derivative of the function \(y = \cos x\) is \(y^{\prime\prime} = -\cos x\).

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Most popular questions from this chapter

Graphing \(f\) and \(f^{\prime}\) a. Graph \(f\) with a graphing utility. b. Compute and graph \(f^{\prime}\) c. Verify that the zeros of \(f^{\prime}\) correspond to points at which \(f\) has \(a\) horizontal tangent line. $$f(x)=\left(x^{2}-1\right) \sin ^{-1} x \text { on }[-1,1]$$

Find the slope of the curve \(5 \sqrt{x}-10 \sqrt{y}=\sin x\) at the point \((4 \pi, \pi)\).

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. $$f(x)=3 x-4$$

The following limits equal the derivative of a function \(f\) at a point a. a. Find one possible \(f\) and \(a\) b. Evaluate the limit. $$\lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{6}+h\right)-\frac{1}{2}}{h}$$

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. $$f(x)=\sqrt{x+2}, \text { for } x \geq-2$$
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