How to find the derivative of a function of a function. Please give an example.

To determine the derivative of a function of a function,
we'll have to apply the chain rule.

If u is a function of v
and v is a function of x, then we can say that u is the function of the function
v.

We'll write in this
manner:

du/dx =
(du/dv)*(dv/dx) 

The first step is to find the function v
(usually is the function inside the brackets or the argument of the trigonometric
functions, or the argument of the logarithmic functions, or the expression under the
square root, etc.)

Then, we'll re-write the function u in
terms of v and we'll differentiate u with respect to
v.

We'll re-write the results with respect to
x.

Example:

u = (x^2 +
5x)^3

We'll have to differentiate u with respect to
x:

du/dx = (d/dx)(x^2 +
5x)^3

We'll substitute the expression inside the brackets
by v.

v = x^2 + 5x

u =
v^3

To apply the chain rule, we'll have to differentiate u
with respect to v:

du/dv =
(v^3)'

du/dv = 3v^2

Now, we'll
differentiate v with respect to x:

dv/dx = (x^2 +
5x)'

dv/dx = 2x + 5

du/dx =
(du/dv)(dv/dx)

du/dx = 3[(x^2 + 5x)^2]*(2x +
5)