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Chapter 2: Q. 38 (page 233)

Find the derivatives of each of the functions in Exercises 17鈥�50. In some cases it may be convenient to do some preliminary algebra

f(x)=3sin-1x2

Short Answer

Expert verified

The derivative of the function is6sin-1x1-x2.

Step by step solution

01

Step 1. Given Information.

The given function isf(x)=3sin-1x2.

02

Step 2. Preliminary Algebra.

We know,

(fg)'(x)=f'(x)g(x)+f(x)g'(x)fg'(x)=f'(x)g(x)-f(x)g'(x)(g(x))2(f鈭�u)'(x)=f'(u(x))u'(x)

03

Step 3. Derivative of the function.

The derivative of the function will be,

f'(x)=ddx3sin-1x2=32sin-1xddxsin-1x[Using chain rule]=32sin-1x11-x2=6sin-1x1-x2

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Most popular questions from this chapter

Use the definition of the derivative to find ffor each function fin Exercises 39-54

f(x)=1x2-1

Find a function that has the given derivative and value. In each case you can find the answer with an educated guess and check process it may be helpful to do some preliminary algebra

f'(x)=x(4-2x);f(0)=0

Taking the limit: We have seen that if f is a smooth function, then f'(c)鈮�f(c+h)-f(c)hThis approximation should get better as h gets closer to zero. In fact, in the next section we will define the derivative in terms of such a limit.

f'(c)=limh鈫�0f(c+h)-f(c)h.

Use the limit just defined to calculate the exact slope of the tangent line tof(x)=x2atx=4.

Differentiate f(x)=3x+x2in three ways. When you have completed all three parts, show that your three answers are the same:

(a) with the chain rule

(b) with the product rule but not the chain rule

(c) without the chain or product rules.

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=x+45
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