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Chapter 3: Problem 31

Verify the following derivative formulas using the Quotient Rule. $$\frac{d}{d x}(\csc x)=-\csc x \cot x$$

Short Answer

Expert verified
Question: Verify the derivative formula for the cosecant function using the Quotient Rule: $$\frac{d}{d x}(\csc x)=-\csc x \cot x$$

Step by step solution

01

Express the cosecant function as a reciprocal

The cosecant function can be expressed as the reciprocal of the sine function: $$\csc x = \frac{1}{\sin x}$$
02

Apply the Quotient Rule

The Quotient Rule states that for a function \(g(x) = \frac{F(x)}{G(x)}\), its derivative \(g'(x) = \frac{F'(x) G(x) - F(x) G'(x)}{[G(x)]^2}\). So applying the Quotient Rule on \(\csc x = \frac{1}{\sin x}\): $$\frac{d}{d x}(\csc x) = \frac{d}{d x} \left(\frac{1}{\sin x}\right) = \frac{-(\cos x)(1) - (0)(\sin x)}{(\sin x)^2}$$
03

Simplify the result

The expression for the derivative simplifies to: $$\frac{d}{d x}(\csc x) = \frac{-\cos x}{(\sin x)^2}$$
04

Rewrite the result using trigonometric functions

We can rewrite the simplified expression using the definition of the cotangent function: $$\frac{d}{d x}(\csc x) = -\csc x \cot x$$ Now, we have verified the given derivative formula for the cosecant function using the Quotient Rule.

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Most popular questions from this chapter

Savings plan Beginning at age \(30,\) a self-employed plumber saves \(\$ 250\) per month in a retirement account until he reaches age \(65 .\) The account offers \(6 \%\) interest, compounded monthly. The balance in the account after \(t\) years is given by \(A(t)=50,000\left(1.005^{12 t}-1\right)\) a. Compute the balance in the account after \(5,15,25,\) and 35 years. What is the average rate of change in the value of the account over the intervals \([5,15],[15,25],\) and [25,35]\(?\) b. Suppose the plumber started saving at age 25 instead of age 30\. Find the balance at age 65 (after 40 years of investing). c. Use the derivative \(d A / d t\) to explain the surprising result in part (b) and to explain this advice: Start saving for retirement as early as possible.

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