How to Use Implicit Differentiation to Find dy/dx given y^2 + xcosh(y) + sinh^2(x) = 50

TL;DR

Learn how to find d y d x using implicit differentiation, even when it's difficult or impossible to solve for y.

Transcript

this problem we have this equation and we're asked to find d y d x so we're going to use what's called implicit differentiation because we have a y squared here we have a y inside the cosine so it's going to be pretty hard if not impossible to solve for y so in order to find the derivative what we'll do is we'll just take the derivative of both sid... Read More

Key Insights

• 😀 Implicit differentiation is helpful when solving for y is difficult or not possible.
• 📏 The chain rule and product rule are important tools in differentiating in implicit differentiation.
• ➗ The final step involves dividing the derivatives by the common factor to find d y d x.

Questions & Answers

Q: What is implicit differentiation used for?

Implicit differentiation is used when it is difficult or impossible to solve for y in an equation, allowing us to find the derivative d y d x.

Q: What is the chain rule and how is it applied in implicit differentiation?

The chain rule is used when differentiating functions within functions. In implicit differentiation, when y is a function of x, we treat it as a separate function and use the chain rule to differentiate it.

Q: How does product rule apply in implicit differentiation?

The product rule is used when differentiating the product of two functions. In implicit differentiation, when there are multiple terms, we apply the product rule to differentiate those terms.

Q: How do we find d y d x in the final step of implicit differentiation?

In the final step, we divide the derivatives of the individual terms by the common factor (2y + x cinch y) to find d y d x.

Summary & Key Takeaways

• Implicit differentiation is used to find the derivative when it is hard to solve for y in the given equation.

• The process involves taking the derivative of both sides of the equation with respect to x.

• The chain rule and product rule are used to simplify and differentiate the equation.