Question

What is the derivative of ln ln 4x ))?

Answer 1

Refer to the image attached.

Answer 2

The derivative of ln(ln(4x)) with respect to x is found using the chain rule and is 1/(xln(4x)).

Step-by-step explanation:

The student has asked for the derivative of the function ln(ln(4x)). To find the derivative, we use the chain rule. First, let's denote the inner function as u = ln(4x), so our function becomes ln(u).

Applying the chain rule:

1. Find the derivative of the outer function with respect to u, which is 1/u since the derivative of ln(u) is 1/u.
2. Find the derivative of the inner function with respect to x, which by using the property that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number, we get 1/(4x) × 4 = 1/x.
3. Multiply the derivatives from steps 1 and 2 together: (1/u) × (1/x) = 1/(xln(4x)).

Therefore, the derivative of ln(ln(4x)) with respect to x is 1/(xln(4x)).