Question

1-22: Differentiate.

2. \(f\left( x \right) = \tan x - 4\sin x\)

Short answer

Therequired value is \(f'\left( x \right) = {\sec ^2}x - 4\cos x\).

Step-by-step solution

Write the formula of the derivatives of trigonometric functions 

\(\begin{aligned}\frac{d}{{dx}}\left( {\tan x} \right) &= {\sec ^2}x\\\frac{d}{{dx}}\left( {\sin x} \right) &= \cos x\end{aligned}\)

Find the differentiation of the function

Consider the function \(f\left( x \right) = \tan x - 4\sin x\). Differentiate the function w.r.t \(x\) by using the derivatives of trigonometric functions.

\(\begin{aligned}\frac{{d\left( {f\left( x \right)} \right)}}{{dx}} &= \frac{{d\left( {\tan x - 4\sin x} \right)}}{{dx}}\\ &= \frac{d}{{dx}}\left( {\tan x} \right) - 4\frac{d}{{dx}}\left( {\sin x} \right)\\ &= {\sec ^2}x - 4\cos x\end{aligned}\)

Thus, the derivative of the function \(f\left( x \right) = \tan x - 4\sin x\) is \(f'\left( x \right) = {\sec ^2}x - 4\cos x\).