# Find the equation of the tangent line for the given curve.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 3.4, Problem 42E
To determine

## Find the equation of the tangent line for the given curve.

Expert Solution

The equation of the tangent line is y=2x1 .

### Explanation of Solution

Given:

The given curve is y=1+x3 and the given point is (2,3) .

Calculation:

Find the slope of the tangent line at the given point (2,3) .

Find first derivative of the given equation y=1+x3 .

Apply chain rule.

Let f=a12,a=1+x3

y'=dda(a12)ddx(1+x3)

Use derivative rule ddx(xn)=nxn1 .

y'=3x22a

Substitute the value of a=1+x3 .

y'=3x221+x3

Plug in the x=2 and y=3 into the derivative.

y'=32221+23=126=2

Slope of the tangent line is 2 .

Use point-slope form of the equation (For the equation of tangent line).

yy1=m(xx1)y1=3,x1=2,m=2y3=2(x2)y3=2x4y3+3=2x4+3y=2x1

Hence theequation of the tangent line is y=2x1 .

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