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

f(x)=1-2x

Short Answer

Expert verified

The derivative of the function isf'(x)=2,ifx>12DNE,ifx=12-2.ifx<12.

Step by step solution

01

Step 1. Given Information

The given function isf(x)=1-2x.

02

Step 2. Rewrite the function

The given function can be rewritten as, f(x)=-(1-2x),ifx121-2x,ifx>12.

03

Step 3. Find the derivative

  • It is known that, the derivative of the linear function is (mx+b)'=m.
  • Find the derivative for x>12.

f'(x)=ddx(-(1-2x))=ddx(2x-1)=2

  • Find the derivative for x>12.

f'(x)=ddx(1-2x)=-2

  • Since 2-2, the derivatives at left and right pieces are not equal at x=12. The derivative does not exists at x=12.
  • So, the derivative of the function is, f'(x)=2,ifx>12DNE,ifx=12-2.ifx<12.

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