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Chapter 2: Q. 42 (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)=sinsin-1x.

Short Answer

Expert verified

The derivative is1.

Step by step solution

01

Step 1. Given Information.

The given function isf(x)=sinsin-1x.

02

Step 2. Preliminary Algebra.

We know,

ddx(sinx)=cosxddxsin-1x=11-x2(fg)'(x)=f'(x)g(x)+f(x)g'(x)

03

Step 3. Derivative of the function.

The derivative of the function will be,

ddxsinsin-1x=ddx(x)Since,sin-1xis the inverse ofsinxTherefore,ddxsinsin-1x=1

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Most popular questions from this chapter

Differentiation review: Without using the chain rule find the derivative of each of the function f that follows some algebra may be required before differentiating

a)f(x)=(3x+1)4b)f(x)=(1x+1x2)2c)f(x)=(x+1)2xd)f(x)=(x+1)2x

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)=x2+1

For each function graphed in Exercises 65-68, determine the values of xat which ffails to be continuous and/or differentiable. At such points, determine any left or right continuity or differentiability. Sketch secant lines supporting your answers.

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.

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)=3x5-2x2+4;f(0)=1
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