   Chapter 3.1, Problem 40E

Chapter
Section
Textbook Problem

Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen. y = x − x , ( 1 , 0 )

To determine

To find: The equation of the tangent line to the curve and to sketch the given curve and the tangent line at the given point.

Explanation

Given:

The equation of the curve is y=xx and the point is (1,0).

Derivative rules:

(1) Power rule: ddx(xn)=nxn1

(2) Difference rule:ddx[f(x)g(x)]=ddx(f(x))ddx(g(x))

Formula used:

The equation of the tangent line at (x1,y1) is, yy1=m(xx1) (1)

where, m is the slope of the tangent line at (x1,y1) and m=dydx|x=x1.

Calculation:

The derivative of y is dydx , which is obtained as follows.

dydx=ddx(y) =ddx(xx) =ddx(xx12)

Apply the difference rule (2),

dydx=ddx(x)ddx(x12)

Simplify the expression using the power rule (1),

dydx=(1x11)(12x121)  =(1x0

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