Implicit derivative of y = cos(5x - 3y) | Taking derivatives | Differential Calculus | Khan Academy | Summary and Q&A

TL;DR

This video explains how to find the rate of change of a function using implicit differentiation.

Key Insights

• 😀 Implicit differentiation is used when a function cannot be easily expressed as y=f(x).
• 📏 The chain rule is a crucial tool in differentiating composite functions.
• 😑 Implicit differentiation allows for finding the derivative of a function even if it is not expressed explicitly.
• 🍉 Solving implicit differentiation problems often involves algebraic manipulation and rearrangement of terms.

Transcript

Let's say we have the relationship y is equal to cosine of 5x minus 3y. And what I want to find is the rate at which y is changing with respect to x. And we'll assume that y is a function of x. So let's do what we've always been doing. Let's apply the derivative operator to both sides of this equation. On the left-hand side, right over here, we get... Read More

Questions & Answers

Q: What does the derivative of the cosine function with respect to its argument equal?

The derivative of the cosine function with respect to its argument is equal to negative sine of that argument.

Q: How do you differentiate the term -3y with respect to x?

To differentiate -3y with respect to x, you multiply the constant -3 by the derivative dy/dx of y with respect to x.

Q: What is the general approach for solving implicit differentiation problems?

The general approach for solving implicit differentiation problems is to apply the derivative operator to both sides of the equation and then use algebraic manipulation to isolate the derivative of interest.

Q: How do you find dy/dx in implicit differentiation?

To find dy/dx in implicit differentiation, algebraically rearrange the equation to isolate dy/dx on one side and then divide by the remaining terms.

Summary & Key Takeaways

• The video demonstrates how to find the rate of change of a function by applying the derivative operator to both sides of an equation.

• The chain rule is then used to differentiate the cosine function, and the derivatives of the variables are computed.

• By rearranging the equation and solving for the rate of change, the video concludes with the expression dy/dx.