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Use the differentiation rules developed in this section to find the derivatives of the functions in Exercises 35-64. Note that it may be necessary to do some preliminary algebra before differentiating.

f(x)=(x-2)2(x2+1)(x-3)

Short Answer

Expert verified

The derivative is(2x-4)(x3-3x2+x-3)-(x2+4-4x)(3x2-6x+1)(x3-3x2+x-3)2.

Step by step solution

01

Step 1. Given Information

The given function isf(x)=(x-2)2(x2+1)(x-3).

02

Step 2. Simplify the function

Simplify the function.

f(x)=x2+4-4xx2(x-3)+(x-3)=x2+4-4xx3-3x2+x-3

03

Step 3. Find the derivative

Apply the quotient rule of derivative, fg'(x)=f'(x)g(x)-f(x)g'(x)(g(x))2.

f'(x)=ddx(x2+4-4x)×(x3-3x2+x-3)-(x2+4-4x)ddx(x3-3x2+x-3)(x3-3x2+x-3)2=(2x-4)(x3-3x2+x-3)-(x2+4-4x)(3x2-6x+1)(x3-3x2+x-3)2

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