Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

4. Applications of Derivatives

Implicit Differentiation

Calculus

4. Applications of Derivatives

Implicit Differentiation

Previous problemNext problem

3:08 minutes

Problem 3.8.13b

Textbook Question

13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.x⁴+y⁴ = 2;(1,−1)

Verified step by step guidance

Start by differentiating both sides of the equation with respect to x. The equation is x⁴ + y⁴ = 2. Use implicit differentiation, which involves differentiating y with respect to x as well.

Differentiate x⁴ with respect to x, which gives 4x³. For y⁴, apply the chain rule: differentiate y⁴ with respect to y to get 4y³, then multiply by dy/dx (the derivative of y with respect to x).

Set up the equation from the differentiation: 4x³ + 4y³(dy/dx) = 0. This equation represents the derivative of the original equation.

Solve for dy/dx, which represents the slope of the curve. Rearrange the equation to isolate dy/dx: dy/dx = -4x³ / 4y³.

Substitute the given point (1, -1) into the equation for dy/dx. Replace x with 1 and y with -1 to find the slope at this specific point.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

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 curves defined by equations like x⁴ + y⁴ = 2, where y cannot be easily isolated.

Slope of a Curve

The slope of a curve at a given point represents the rate of change of the y-coordinate with respect to the x-coordinate at that point. Mathematically, it is found by evaluating the derivative of the function at the specified point. In the context of implicit differentiation, once we find dy/dx, we can substitute the coordinates of the point (1, -1) to determine the slope of the curve at that location.

Chain Rule

The chain rule is a fundamental principle in calculus used 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 rule is essential in implicit differentiation, as it allows us to differentiate terms involving y when applying the derivative to both sides of an equation.