find the derivative
F(x)=tan(ln(5x))
To find the derivative of F(x) = tan(ln(5x)), we need to use the chain rule. The chain rule states that if we have a composition of functions, the derivative is found by taking the derivative of the outer function and multiplying it by the derivative of the inner function.
Let's break down the steps to find the derivative of F(x):
Step 1: Identify the outer function and the inner function.
In this case, the outer function is tan(u) and the inner function is ln(5x).
Step 2: Find the derivative of the outer function.
The derivative of tan(u) is sec^2(u).
Step 3: Find the derivative of the inner function.
The derivative of ln(5x) can be found using the chain rule again. The derivative of ln(u) is (1/u) multiplied by the derivative of u.
So, the derivative of ln(5x) is (1/(5x)) multiplied by the derivative of 5x, which is simply 5.
Step 4: Apply the chain rule.
Using the chain rule, we multiply the derivative of the outer function (sec^2(u)) by the derivative of the inner function [(1/(5x)) * 5].
Therefore, the derivative of F(x) = tan(ln(5x)) is:
F'(x) = sec^2(ln(5x)) * (1/(5x)) * 5.
Simplifying this expression will give you the final result for the derivative.