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Chapter 2: Q. 69 (page 198)

Find the derivatives of each of the absolute value and piecewise-defined functions in Exercises 65-72.

f(x)=x3,ifx<1x,ifx鈮�1

Short Answer

Expert verified

The derivative of the function isf'(x)=3x2,ifx<1DNE,ifx=11,ifx>1.

Step by step solution

01

Step 1. Given Information

The given function isf(x)=x3,ifx<1x,ifx鈮�1.

02

Step 2. Find the derivative

  • It is known that, the power rule of derivative is (xn)'=nxn-1.
  • Find the derivative for x<1.

f'(x)=ddx(x3)=3x2

  • Find the derivative for x>1.

f'(x)=ddx(x)=1

  • Since 3(1)2鈮�1, the derivatives at left and right pieces are not equal at x=1. The derivative does not exists at x=1.
  • So, the derivative of the function is, f'(x)=3x2,ifx<1DNE,ifx=11,ifx>1.

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Most popular questions from this chapter

Use the definition of the derivative to find f'for each function fin Exercises 34-59

role="math" localid="1648284617718" f(x)=2x+1

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.

Use the definition of the derivative to prove the following special case of the product rule

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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)=x13-2x-1
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