Question

What is the derivative of `f(x)=log_4(e^x+3)` ?

Answer 1

First, we will rewrite the function in terms of natural logarithms, using the change-of-base rule:

`f(x) = ln(e^x + 3)/ln4`

Differentiating will require use of the chain rule:

`d/dx f(x) = 1/ln 4 * d/(d(e^x + 3))[ln(e^x + 3)] * d/dx[e^x + 3]`

We know that since the derivative of `ln x` with respect to `x` is `1/x`, then the derivative of `ln(e^x + 3)` with respect to `e^x + 3` will be `1/(e^x + 3)`. We also know that the derivative of `e^x + 3` with respect to `x` will simply be `e^x`:

`d/dx f(x) = 1/ln 4 * 1/(e^x + 3) * (e^x)`

Simplifying yields:

`d/dx f(x) = (e^x)/(ln 4(e^x + 3))`