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Chapter 14: Q. 33 (page 906)

In Problems 33鈥�35, find the derivative of each function at the number indicated.

33.f(x)=鈭�4x2+5at3

Short Answer

Expert verified

The derivative off(x)=鈭�4x2+5at3isf'(3)=-24

Step by step solution

01

Step 1. Find f(3)

To find the derivative we have to use the formula,

f鈥�(c)=limx鈫�cf(x)鈭�f(c)x鈭�c

We have been given f(x)=-4x2+5and c=3

Now we have to findf(c), therefore substitute c=3in f(x)to findf(3).

f(x)=鈭�4x2+5f(3)=鈭�4(3)2+5=鈭�36+5=鈭�31

02

Step 2. Use the derivative formula

Substituting all the values in the derivative formula we get,

f鈥�(3)=limx鈫�3f(x)鈭�f(3)x鈭�(3)=limx鈫�3鈭�4x2+5鈭�(鈭�31)x鈭�3=limx鈫�3鈭�4x2+36x鈭�3=limx鈫�3鈭�4(x鈭�3)(x+3)x鈭�3=limx鈫�3(鈭�4x鈭�12)=(鈭�4(3)鈭�12)=鈭�12鈭�12=鈭�24

Hence the derivative of f(x)whenc=3isf鈥�(3)=鈭�24.

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