   Chapter 2.2, Problem 23E

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# Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. f ( x ) = x 2 − 2 x 3

To determine

To find: the derivative of the function by using the definition of derivative. State the domain of function and domain of derivative of its function.

Explanation

Definition of derivative is

f'x= limh0 fx+h-f(x)h

Here fx=x2-2x3

Therefore, fx+h=x+h2-2x+h3

By expanding the square and cube term by using the identities

(a + b)2 = a2+ 2ab +b2 and  (a + b)3 = a3 + 3a2b + 3ab2 + b3   we have,

fx+h=x2+2xh+h2-2(x3+3x2h+3xh2+h3)

By distributing -2 in last term

fx+h=x2+2xh+h2-2x3-6x2h-6xh2-2h3

By substituting these values in the formula

f'x= limh0 x2+2xh+h2-2x3-6x2h-3xh2-2h3-x2-2x3h

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