   Chapter 2.8, Problem 26E

Chapter
Section
Textbook Problem

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. g ( t ) = 1 t

To determine

To find: The derivative of the function g(t)=1t and state the domain of the function and its derivative.

Explanation

Formula used:

The derivative of a function f , denoted by f(x), is

f(x)=limh0f(x+h)f(x)h (1)

Difference of squares formula: (a2b2)=(a+b)(ab)

Calculation:

Obtain the derivative of the function g(t).

Use the equation (1) to compute g(t),

g(t)=limh0g(t+h)g(t)h=limh01t+h1th=limh0tt+htt+hh=limh0(tt+h)htt+h

Multiply both the numerator and the denominator by the conjugate of the numerator,

g(t)=limh0(tt+h)htt+h×(t+t+h)(t+t+h)=limh0(tt+h)(t+t+h)htt+h(t+t+h)

Apply the difference of square formula,

g(t)=limh0(t)2(t+h)2h(tt+ht+t

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