Derivative of triple composition | Taking derivatives | Differential Calculus | Khan Academy

TL;DR

The derivative of G with respect to x at x=5 is equal to 0.

Transcript

Consider the functions f and g with the graphs shown below. If capital G of x is equal to g of f of x whole thing squared, what is the value of G prime, capital G prime, of 5? And I encourage you to now pause this video and try to solve it on your own. So let's try to think through this somewhat complicated-looking function definition right over he... Read More

Key Insights

• ❓ The function G(x) can be defined as the composition of multiple functions.
• 🫡 The chain rule is used to calculate the derivative of G with respect to x.
• 👻 Substituting the specific value of x=5 allows for the determination of G prime of 5.

Questions & Answers

Q: How is G(x) defined?

G(x) is equal to the square of g(f(x)), where g(x) and f(x) are given functions.

Q: How can the derivative of G with respect to x be calculated?

The chain rule can be used to find the derivative of G with respect to x by taking the derivative of the outer function, the derivative of the inner function, and multiplying them together.

Q: What is the value of G prime of 5?

By substituting x=5 into the derivative expression, it is found that G prime of 5 is equal to 0.

Q: Why is G prime of 5 equal to 0?

Since the derivative involves three factors multiplied together, and one of the factors (f prime of x) is equal to 0 at x=5, the entire expression evaluates to 0.

Summary & Key Takeaways

• The function G(x) is defined as the square of g(f(x)), where g(x) and f(x) are given.

• The derivative of G with respect to x can be calculated using the chain rule.

• By substituting x=5 into the derivative expression, it is determined that G prime of 5 is equal to 0.