How would you determine the derivative of \((3x+5)^5\)?

At this stage you may try multiplying out the brackets and then differentiating. However, expanding \((3x+5)^5\) can be a bit challenging. In this section we will be looking at an alternative called the CHAIN RULE. The 2 formats we will be looking at are:

\({d \over dx}[f(x)]^n=n[f(x)]^{n-1}f^{'}(x)\)

\({d\over dx}f[g(x)]=f'[g(x)]g^{'}(x)\)

TRY THIS

Determine \({dy\over dx}\) for each of the following:

1. \((x^3+4)^5\)
2. \((4x^5+2)^3\)
3. \((5x+x^2)^{-{2\over3}}\)
4. \(\sqrt{x^4-9x}\)

SHOW SOLUTION

1. \(15x^2(x^3+4)^4\)
2. \(60x^4(4x^5+2)^2\)
3. \(-{2\over3}(5+2x)(5x+x^2)^{-{5\over3}}\)
4. \({1\over2}(4x^3-9)(x^4-9x)^{-{1\over2}}\)