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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Find an equation of the tangent line to the given curve at the specified point.

y = x 2 1 x 2 + x + 1 , ( 1 , 0 )

To determine

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

Explanation

Given:

The curve is y=x21x2+x+1.

The point is (1,0).

Derivative rules:

(1) Quotient Rule: If f1(x) and f2(x) are both differentiable, then

ddx[f1(x)f2(x)]=f2(x)ddx[f1(x)]f1(x)ddx[f2(x)][f2x]2

(2) Power Rule: ddx(xn)=nxn1

(3) Sum Rule: ddx[f(x)+g(x)]=ddx(f(x))+ddx(g(x))

(4) 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(x21x2+x+1)

Apply the quotient rule (1) and simplify the terms,

dydx=(x2+x+1)ddt(x21)(x21)ddx(x2+x+1)(x2+x+1)2

Apply the derivative rule (3) and (4),

dydx=(x2+x+1)(ddt(x2)

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