Skip to main content
Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.8.39
Chapter 3, Problem 3.8.39

27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
√x⁴+y² = 5x+2y³

Verified step by step guidance
1
Start by differentiating both sides of the equation with respect to x. Remember that y is a function of x, so when differentiating terms involving y, use the chain rule.
Differentiate the left side: For √(x⁴ + y²), use the chain rule. The derivative of √u with respect to u is 1/(2√u), and then multiply by the derivative of the inside function (x⁴ + y²) with respect to x.
Differentiate the right side: For 5x + 2y³, differentiate each term separately. The derivative of 5x with respect to x is 5, and for 2y³, use the chain rule: differentiate y³ with respect to y, which is 3y², and then multiply by dy/dx.
Set the derivatives from both sides equal to each other. This will give you an equation involving dy/dx.
Solve the resulting equation for dy/dx. This will involve isolating dy/dx on one side of the equation, which may require algebraic manipulation.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
9m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. Instead of solving for one variable in terms of the other, we differentiate both sides of the equation with respect to the independent variable, applying the chain rule as necessary. This method is particularly useful for equations that are difficult or impossible to rearrange.
Recommended video:
05:14
Finding The Implicit Derivative

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. It states that if a function y is defined as a function of u, which in turn is a function of x, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This is essential in implicit differentiation when dealing with terms involving y.
Recommended video:
05:02
Intro to the Chain Rule

Derivative Notation (dy/dx)

The notation dy/dx represents the derivative of y with respect to x, indicating the rate of change of y as x changes. In the context of implicit differentiation, dy/dx is treated as a variable that can be solved for, even when y is not explicitly defined in terms of x. Understanding this notation is crucial for interpreting the results of differentiation and for solving for the slope of the tangent line at a given point.
Recommended video:
04:22
Sigma Notation
Related Practice
Textbook Question

Find d/dx(ln√x²+1).

Textbook Question

Find the derivative of the following functions.

y = e^-x sin x

Textbook Question

47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.

f(x)=tan x; (1,π/4)

Textbook Question

9–61. Evaluate and simplify y'.


y = (2x−3)x^3/2

Textbook Question

Find an equation of the line tangent to the curve y = sin x at x = 0.

1
views
Textbook Question

A 12-ft ladder is leaning against a vertical wall when Jack begins pulling the foot of the ladder away from the wall at a rate of 0.2 ft/s. What is the configuration of the ladder at the instant when the vertical speed of the top of the ladder equals the horizontal speed of the foot of the ladder?