# How do you differentiate f(x)=2sinx-tanx?

Apr 30, 2018

The derivative is $2 C o s \left(x\right) - \left(\frac{1}{C} o {s}^{2} \left(x\right)\right)$- see below for how to do it.

#### Explanation:

If

$f \left(x\right) = 2 S \in x - T a n \left(x\right)$

For the sine part of the function, the derivative is simply: $2 C o s \left(x\right)$

However, $T a n \left(x\right)$ is a bit more tricky- you have to use the quotient rule.

Recall that $T a n \left(x\right) = \left(S \in \frac{x}{C} o s \left(x\right)\right)$

Hence we can use The quotient rule

if$f \left(x\right) = \left(S \in \frac{x}{C} o s \left(x\right)\right)$

Then

$f ' \left(x\right) = \left(\frac{C o {s}^{2} \left(x\right) - \left(- S {\in}^{2} \left(x\right)\right)}{C o {s}^{2} \left(x\right)}\right)$

$S {\in}^{2} \left(x\right) + C o {s}^{2} \left(x\right) = 1$

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

So the complete function becomes
$f ' \left(x\right) = 2 C o s \left(x\right) - \left(\frac{1}{C} o {s}^{2} \left(x\right)\right)$

Or

$f ' \left(x\right) = 2 C o s \left(x\right) - S e {c}^{2} \left(x\right)$

Apr 30, 2018

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

#### Explanation:

$\text{utilising the "color(blue)"standard derivatives}$

•color(white)(x)d/dx(sinx)=cosx" and "d/dx(tanx)=sec^2x

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