   Chapter 3.5, Problem 30E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Use implicit differentiation to find an equation of the tangent line to the curve at the given point.30. x 2 / 3 + y 2 / 3 =   4 ,   ( − 3 3 , 1 ) , (astroid) To determine

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

Explanation

Given:

The curve is x23+y23=4.

The point is (33,1).

Derivative rules: Chain rule

If y=f(u) and u=g(x)  are both differentiable functions, then dydx=dydududx.

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,y=y1.

Calculation:

Consider the equation x23+y23=4.

Differentiate the given equation implicitly with respect to x,

ddx(x23+y23)=ddx(4)ddx(x23)+ddx(y23)=023x231+ddx(y23)=023x13+ddx(y23)=0

Apply the chain rule and simplify the terms,

23x13+[ddy(y23)dydx]=023x13+[23y13dydx]=023x13+23y13dydx=023y13dydx=23x13

Separate dydx to one side of the equation,

23y13dyd

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### Convert the expressions in Exercises 6584 to power form. 3x453x+43xx

Finite Mathematics and Applied Calculus (MindTap Course List)

#### In Exercises 47-52, find and simplify f(a+h)f(a)h(h0) for each function. 51. f(x)=1x

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

#### Evaluate in problems 1-4. Write all the answers without using exponents. 2. (a) ...

Mathematical Applications for the Management, Life, and Social Sciences

#### For y = x3 ln x, y = a) x2(1 + 3 ln x) b) x3 + 3x2 (ln x) c) 3x d) 4x2(ln x)

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 