Question

1-22: Differentiate.

1. \(f\left( x \right) = 3\sin x - 2\cos x\)

Short answer

The required value is \(f'\left( x \right) = 3\cos x + 2\sin x\).

Step-by-step solution

Write the formula of the derivatives of trigonometric functions

\(\begin{aligned}\frac{d}{{dx}}\left( {\sin x} \right) &= \cos x\\\frac{d}{{dx}}\left( {\cos x} \right) &= - \sin x\end{aligned}\)

Find the differentiation of the function

Consider the function \(f\left( x \right) = 3\sin x - 2\cos x\). Differentiate the function w.r.t \(x\) by using the derivatives of trigonometric functions.

\(\begin{aligned}\frac{{d\left( {f\left( x \right)} \right)}}{{dx}} &= \frac{{d\left( {3\sin x - 2\cos x} \right)}}{{dx}}\\ &= \frac{d}{{dx}}\left( {3\sin x} \right) + \frac{d}{{dx}}\left( { - 2\cos x} \right)\\ &= 3\left( {\cos x} \right) + \left( { - 2} \right)\left( { - \sin x} \right)\\ &= 3\cos x + 2\sin x\end{aligned}\)

Thus, the derivative of the function \(f\left( x \right) = 3\sin x - 2\cos x\) is \(f'\left( x \right) = 3\cos x + 2\sin x\).