Answer
1) Rewrite $\sec x$ into $\frac{1}{cos x}$
2) Then apply the Quotient Rule for $\frac{d}{dx}(\frac{1}{cos x})$
3) Use trigonometrical rules to simplify.
Work Step by Step
We know that $\sec x=\frac{1}{\cos x}$. So, $$\frac{d}{dx}(\sec x)=\frac{d}{dx}\Bigg(\frac{1}{\cos x}\Bigg)$$
Now apply the Quotient Rule, we have $$\frac{d}{dx}(\sec x)=\frac{(1)'\cos x-1(\cos x)'}{(\cos x)^2}$$
$$\frac{d}{dx}(\sec x)=\frac{0\times\cos x-(-\sin x)}{\cos^2 x}$$
$$\frac{d}{dx}(\sec x)=\frac{\sin x}{\cos^2 x}$$
$$\frac{d}{dx}(\sec x)=\frac{\sin x}{\cos x}\times\frac{1}{\cos x}$$
$$\frac{d}{dx}(\sec x)=\tan x\sec x$$
The statement has been proved.
