Find f'(x) given that d/dx(f(2x)) = x^2

TL;DR

Derivative problem solved using the chain rule and variable substitution, resulting in F prime of x = x^2/8.

Transcript

in this video we're going to solve this problem find F Prime of x if it is known that the derivative with respect to X of f of 2x is equal to x squared let's go ahead and try to work this out solution so we're told that the derivative of f of 2x is equal to x squared so the natural thing to do is to compute that derivative so what we're going to us... Read More

Key Insights

• 💻 The chain rule is an essential concept in calculus for computing derivatives of composite functions.
• 😑 Variable substitution simplifies complex expressions and aids in finding derivatives.
• 🗂️ Dividing by constants helps in isolating and solving for the derivative of a function.

Questions & Answers

Q: How is the chain rule used in computing the derivative of f of 2x?

The chain rule is applied by treating f as the outside function and 2x as the inside function, leading to F Prime of 2x = x squared.

Q: Why is a variable substitution made in the problem?

A variable substitution is made to simplify the expression and transform it into a function of a single variable, making it easier to compute the derivative as F Prime of x = x squared over 8.

Q: What is the significance of dividing by 2 in the solution?

Dividing by 2 in the solution helps to isolate the derivative of f and accurately compute it as F Prime of x = x squared over 2, which is then further simplified using variable substitution.

Q: How does the solution validate the correctness of the derivative computation?

By substituting U back with X and recalculating the expression, one can verify the correctness of the derivative computation as F Prime of x = x squared over 8.

Summary & Key Takeaways

• Using the chain rule, the derivative of f of 2x is computed as x squared.

• To find the derivative of f, the equation is divided by 2, leading to F Prime of 2x = x squared over 2.

• Making a variable substitution of U for 2x, the final derivative is calculated as F Prime of x = x squared over 8.