# 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 44E
To determine

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

Expert Solution

The equation of the tangent line is y=x .

### Explanation of Solution

Given:

The given curve is y=sinx+sin2x and the given point is (0,0) .

Calculation:

Find the slope of the tangent line at the given point (0,0) .

Find first derivative of the given equation y=sinx+sin2x .

Use sum rule.

(f+g)'=f'+g'

y'=ddx(sinx)+ddx(sin2x)

Apply chain rule.

Let f=a2,a=sinx

y'=ddx(sinx)+dda(a2)ddx(sinx)

Use derivative rule ddx(sinx)=cosx and ddx(xn)=nxn1 .

y'=cosx+2acosx

Substitute the value of a=sinx .

y'=cosx+2sinxcosxy'=cosx(1+2sinx)

Plug in the x=0 into the derivative.

y'=cos(0)(1+2sin(0))=1(1+20)=1

Slope of the tangent line is 1 .

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

yy1=m(xx1)y1=0,x1=0,m=1y0=1(x0)y=x

Hence theequation of the tangent line is y=x .

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