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

$f\left(g\right)=g^k$

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

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

Task!

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:

   Answer

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

2. Use the function of a function approach again:

   Answer

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

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

   Answer

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

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

Exercise

Obtain the derivatives of the following functions:

1. ${\left(\frac{2x+1}{3x-1}\right)}^4$
2. $tan\left(3x^2+2x\right)$
3. ${sin}^2\left(3x^2-1\right)$

Answer

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