James Stewart

ISBN: 9781305270343

Textbook Problem

If g(x) = x⁴ – 2, find g'(l) and use it to find an equation of the tangent line to the curve y = x⁴ – 2 at the point (1,–1).

To determine

To find: The derivative of the function g(x)=x4−2 at a=1 and obtain the equation of the tangent line to the curve y=x4−2 at the point (1, −1).

Formula used:

The derivative of a function g at a number a, denoted by g′(a) is,

g′(a)=limx→ag(x)−g(a)x−a (1)

The equation of the tangent line to the curve y=g(x) at the point (a,g(a)) is,

y−g(a)=g′(a)(x−a) (2)

Difference of squares formula: (a2−b2)=(a+b)(a−b)

Calculation:

Obtain the derivative of the function g(x) at a=1.

Compute g′(1) by using the equation (1).

g′(1)=limx→1g(x)−g(1)x−1=limx→1(x4−2)−((1)4−2)x−1=limx→1(x4−2)−(1−2)x−1=limx→1x4−2+1x−1

=limx→1x4−1x−1=limx→1(x2)2−(1)2x−1

Apply the difference of squares formula,

g′(1)=limx→<

Check out a sample textbook solution.

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MathCalculusSingle Variable Calculus: Early Transcendentals, Volume I8th Edition

Single Variable Calculus: Early Tr...

Solutions

Single Variable Calculus: Early Tr...

If g(x) = x4 – 2, find g'(l) and use it to find an equation of the tangent line to the curve y = x4 – 2 at the point (1,–1).

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If g(x) = x⁴ – 2, find g'(l) and use it to find an equation of the tangent line to the curve y = x⁴ – 2 at the point (1,–1).