Textbook Question
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
f(v) = v¹⁰⁰+e^v+10
1
views
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.8.37
27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
6x³+7y³ = 13xy
Verified step by step guidance1
Start by differentiating both sides of the equation with respect to x. The equation is 6x³ + 7y³ = 13xy.
Differentiate the left side: The derivative of 6x³ with respect to x is 18x². For 7y³, use the chain rule: the derivative is 21y²(dy/dx).
Differentiate the right side: Use the product rule for 13xy. The derivative is 13(dy/dx)x + 13y.
Set up the equation from the derivatives: 18x² + 21y²(dy/dx) = 13(dy/dx)x + 13y.
Solve for dy/dx: Rearrange the equation to isolate dy/dx on one side. This involves factoring out dy/dx and simplifying the expression.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
7mWas 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, where we often encounter terms involving both x and y.
Recommended video:
05:02Intro 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, finding dy/dx involves isolating the derivative of y in terms of x and potentially other constants. Understanding this notation is crucial for interpreting the results of differentiation and applying them to problems involving rates of change.
Recommended video:
04:22Sigma Notation
Related Practice
Textbook Question
67–78. Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function.
f(x) = e^3x+1
Textbook Question
Derivative calculations Evaluate the derivative of the following functions at the given point.
f(s) = 2√s-1; a=25
1
views
Textbook Question
Simplify the expression e^xln(x²+1).
4
views
Textbook Question
Tangent lines Find an equation of the line tangent to the graph of f at the given point.
f(x) = sec−1(ex); (ln 2,π/3)
Textbook Question
51–56. Second derivatives Find d²y/dx².
sin x + x²y =10