Solution assignment 14 Product and Quotient rule

Differentiate:

$y=\tan(x)$

This function is not in the list of standard functions. However, from trigonometry we know:

$\tan(x)=\displaystyle\frac{\sin(x)}{\cos)x)}$

and thus we can apply the quotient rule with:

$f(x)=\sin(x)$

and:

$g(x)=\cos(x)$

and thus:

$f'(x)=\cos(x)$

and:

$g'(x)=\sin(x)$

and thus the quotient rule yields:

$y'=\displaystyle\frac{\cos(x)\cos(x)-\sin(x)\cdot-\sin(x)}{\cos^2(x)}=\displaystyle\frac{\cos^2(x)+\sin^2(x)}{\cos^2(x)}$

We can simplify this formula in two different ways. Because:

$\cos^2(x)+\sin^2(x)=1$

we can write:

$y'=\displaystyle\frac{1}{\cos^2(x)}$

Or, when we rewrite the original function we get the following form:

$y'=\displaystyle\frac{\cos^2(x)}{\cos^2(x)}+\displaystyle\frac{\sin^2(x)}{\cos^2(x)}=1+(\displaystyle\frac{\sin(x)}{\cos(x)})^2=1+\tan^2(x)$