Derivative of e_____cos(e_) | Advanced derivatives | AP Calculus AB | Khan Academy

TL;DR

This video explains how to find the derivative of a complex expression involving the product of cosine and e^x, using the chain rule and product rule.

Transcript

Let's now use what we know about the chain rule and the product rule to take the derivative of an even weirder expression. So, we're gonna take the derivative, we're gonna take the derivative of either the cosine of x times the cosine of e to the x. So, let's take the derivative of this. So, we can view this as the product of two functions. So, the... Read More

Key Insights

• 👻 The product rule allows us to find the derivative of the product of two functions.
• 📏 The chain rule is used to find the derivative of composite functions.
• ✖️ The derivative of e^cos(x) is e^cos(x) multiplied by -sin(x).
• ✖️ The derivative of cos(e^x) is -sin(e^x) multiplied by e^x.
• 😑 The final derivative expression can be obtained by substituting the derivative expressions back into the original equation.
• 📏 The process involves multiple applications of the chain rule and product rule.
• 📏 Understanding the chain rule and product rule is essential for finding derivatives of complex expressions.

Questions & Answers

Q: What is the product rule in calculus?

The product rule states that for two functions u(x) and v(x), the derivative of their product is given by u'(x)v(x) + u(x)v'(x).

Q: How is the chain rule used in this example?

The chain rule is used to find the derivatives of e^cos(x) and cos(e^x). It involves finding the derivative of the outer function and multiplying it by the derivative of the inner function.

Q: What is the derivative of e^cos(x)?

The derivative of e^cos(x) is e^cos(x) multiplied by the derivative of cos(x), which is -sin(x).

Q: What is the derivative of cos(e^x)?

The derivative of cos(e^x) is -sin(e^x) multiplied by the derivative of e^x, which is e^x.

Summary & Key Takeaways

• The video explains how to find the derivative of the expression cos(x)*cos(e^x) using the product rule.

• It demonstrates how to use the chain rule to find the derivatives of e^cos(x) and cos(e^x).

• The final result is obtained by substituting the derivative expressions back into the original equation.