### 3 Power functions

An example of a function of a function which often occurs is the so-called power function ${\left[g\left(x\right)\right]}^{k}$ where $k$ is any rational number. This is an example of a function of a function in which

$\phantom{\rule{2em}{0ex}}f\left(g\right)={g}^{k}$

Thus, using the chain rule: if $\phantom{\rule{1em}{0ex}}y={\left[g\left(x\right)\right]}^{k}$ then $\frac{dy}{dx}=\frac{df}{dg}\cdot \frac{dg}{dx}=k\phantom{\rule{0.3em}{0ex}}{g}^{k-1}\frac{dg}{dx}.$

For example, if $y={\left(sinx+cosx\right)}^{1∕3}$ then $\frac{dy}{dx}=\frac{1}{3}{\left(sinx+cosx\right)}^{-2∕3}\left(cosx-sinx\right)$ .

Find the derivatives of the following power functions

1. $y={sin}^{3}x$
2. $y={\left({x}^{2}+1\right)}^{1∕2}$
3. $y={\left({e}^{3x}\right)}^{7}$
1. Note that ${sin}^{3}x$ is the conventional way of writing ${\left(sinx\right)}^{3}$ . Now find its derivative:

$\frac{dy}{dx}=3{\left(sinx\right)}^{2}cosx$ which we would normally write as $3{sin}^{2}xcosx$

2. Use the function of a function approach again:

$\frac{dy}{dx}=\frac{1}{2}{\left({x}^{2}+1\right)}^{-1∕2}2x=\frac{x}{\sqrt{{x}^{2}+1}}$

3. Use the function of a function approach first, and then look for a quicker way in this case:

$\frac{dy}{dx}=7{\left({e}^{3x}\right)}^{6}\left(3{e}^{3x}\right)=21{\left({e}^{3x}\right)}^{7}=21{e}^{21x}$

Note that ${\left({e}^{3x}\right)}^{7}={e}^{21x}\phantom{\rule{2em}{0ex}}\therefore \phantom{\rule{2em}{0ex}}\frac{dy}{dx}=21{e}^{21x}$ directly - a much quicker way.

##### Exercise

Obtain the derivatives of the following functions:

1. ${\left(\frac{2x+1}{3x-1}\right)}^{4}$
2. $tan\left(3{x}^{2}+2x\right)$
3. ${sin}^{2}\left(3{x}^{2}-1\right)$
1. $-\frac{20{\left(2x+1\right)}^{3}}{{\left(3x-1\right)}^{5}}$
2. $2\left(3x+1\right){sec}^{2}\left(3{x}^{2}+2x\right)$
3. $6xsin\left(6{x}^{2}-2\right)$ (remember $sin2x\equiv 2sinxcosx$ )