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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

(a) If f ( x ) = 5 x x .

(b) Find equations of the tangent lines to the curve y = 5 x x at the points (1, 2) and (4, 4).

(c) Illustrate part (b) by graphing the curve and tangent lines on the same screen.

(d) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f′.

(a)

To determine

To find: The derivative of f(x).

Explanation

Given:

The function is f(x)=x5x.

Result used: Chain Rule

If g is differentiable at t and f is differentiable at g(t), then the composite function F=fg defined by F(t)=f(g(t)) is differentiable at x and F is given by the product

F(t)=f(g(t))g(t) (1)

Calculation:

Obtain the derivative of f(x).

f(x)=ddx(f(x))=ddx(x5x)=xddx(5x)+5xddx(x)=xddx(5x)12+5x(1)

Simplify f

(b)

To determine

To find: The equation of the tangent line to the curve at the points.

(c)

To determine

To sketch: The graph of the curve and the tangent lines.

(d)

To determine

To check: The derivative f(x) is reasonable by comparing the graph of f(x) and f(x).

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