Question

Suppose you wish to differentiate$g\left(x\right)={\sin }^2\left(x\right)+{\cos }^2\left(x\right)$.What is the fastest way to do this, and why?

Short answer

The fastest way is to use chain rule and derivates of trigonometric functions.

Step-by-step solution

Step 1. Given Information.

The given function is$g\left(x\right)={\sin }^2\left(x\right)+{\cos }^2\left(x\right)$.

Step 2. Derivative.

The easiest way to find the derivative is using chain rule and derivative of trigonometric functions.

That is,

$$\begin{gathered} \frac{d}{dx}g\left(x\right)=\frac{d}{dx}\left({\sin }^2\left(x\right)+{\cos }^2\left(x\right)\right) \\ {g}^{\text{'}}\left(x\right)=\frac{d}{dx}{\sin }^2\left(x\right)+\frac{d}{dx}{\cos }^2\left(x\right) \\ {g}^{\text{'}}\left(x\right)=2\sin x\frac{d}{dx}\sin \left(x\right)+2\cos x\frac{d}{dx}\cos \left(x\right) \\ =2\sin x\cos x-2\cos x\sin x \end{gathered}$$