# How do you differentiate f(x)=tanx using the quotient rule?

Nov 7, 2015

$f ' \left(\tan x\right) = {\sec}^{2} x$

#### Explanation:

Using the trig identities, recall that $\tan x$ can be rewritten as $\sin \frac{x}{\cos} x$.
Now differentiate using quotient rule:

$\frac{d}{\mathrm{dx}} \left(f \frac{x}{g} \left(x\right)\right) = \frac{g \left(x\right) \cdot f ' \left(x\right) - f \left(x\right) \cdot g ' \left(x\right)}{g} {\left(x\right)}^{2}$

$f ' \left(x\right) = \frac{\cos x \cdot \cos x - \sin x \cdot - \sin x}{{\cos}^{2} x}$

$f ' \left(x\right) = \frac{{\cos}^{2} x + {\sin}^{2} x}{\cos} ^ 2 x$

Recall from the trig identities that ${\cos}^{2} x + {\sin}^{2} x = 1$. Simplify:

$f ' \left(x\right) = \frac{1}{\cos} ^ 2 x$

Finally, simplify again with trig identities and the result is:

$f ' \left(x\right) = {\sec}^{2} x$