How do I find the derivative of ln(e^(4x)+3x)?

1 Answer
Jun 21, 2016

(f(g(x)))'=(4e^(4x)+3)/(e^(4x)+3x)

Explanation:

We can find the derivative of this function using chain rule that says:

color(blue)((f(g(x)))'=f'(g(x))*g'(x))

Let us decompose the given function into two functions f(x) and g(x) and find their derivatives as follows:

g(x)=e^(4x)+3x
f(x)=ln(x)

Let's find the derivative of g(x)
Knowing the derivative of exponential that says:
(e^(u(x)))'=(u(x))'*e^(u(x))
So,
(e^(4x))'=(4x)'*e^(4x)=4e^(4x)
Then ,
color(blue)(g'(x)=4e^(4x)+3)

Now Lets find f'(x)

f'(x)=1/x
According to the property above we have to find f'(g(x)) so let's substitute x by g(x) in f'(x) we have:

f'(g(x))=1/g(x)
color(blue)(f'(g(x))=1/(e^(4x)+3x))
Therefore,
(f(g(x)))'=(1/(e^(4x)+3x))*(4e^(4x)+3)

color(blue)((f(g(x)))'=(4e^(4x)+3)/(e^(4x)+3x))