Hello,

Here's the question f(x) = (x + 1)(x^2 + 2)(x^3 + 3). Find the derivative.

I know that with 2 expressions such as f(x) = (2x^2 + x - 1)(x^3 + x), to find the derivative it's a matter of getting the derivative of the 1st expression times the 2nd expression PLUS the first expression times the derivative of the 2nd expression.

How do we do this when there are 3 expressions?

There are a couple of ways to do this. One is the brute strength way: combine two of the parenthetical expressions together, then apply the usual rule as you described it:

f(x) = (x+1)(x^2+2)(x^3+3) = (x^3+x^2+2x+2)(x^3+3)

The other way is to apply the rule recursively. If we let a(x) = x+1, b(x) = x^2+2) and c(x) = (x^3+3), then:

f(x) = a(x)*b(x)*c(x)
f'(x) = a'(x)* [b(x)*c(x)] + a(x)*[b'(x)*c(x) + b(x)*c'(x)]

Rearrange:

f'x) = a'(x) b(x) c(x) + a(x) b'(x) c(x) + a(x) b(x) c'(x)

See the pattern?

Hey! I have never thought of that! Cool! Thanks ankara55t for asking and ebaines for replying! :)