Concept:

Suppose that we have two functions f(x) and g(x) and they are both differentiable.

• Chain Rule: \(\frac{{\rm{d}}}{{{\rm{dx}}}}\left[ {{\rm{f}}\left( {{\rm{g}}\left( {\rm{x}} \right)} \right)} \right] = {\rm{\;f'}}\left( {{\rm{g}}\left( {\rm{x}} \right)} \right){\rm{g'}}\left( {\rm{x}} \right)\)
• Product Rule: \(\frac{{\rm{d}}}{{{\rm{dx}}}}\left[ {{\rm{f}}\left( {\rm{x}} \right){\rm{\;g}}\left( {\rm{x}} \right)} \right] = {\rm{\;f'}}\left( {\rm{x}} \right){\rm{\;g}}\left( {\rm{x}} \right) + {\rm{f}}\left( {\rm{x}} \right){\rm{\;g'}}\left( {\rm{x}} \right)\)

Formulas:

\(\rm\frac{d\tan x}{dx} = \sec^2 x\)

\(\rm\frac{d\sec x}{dx} =\sec x \tan x\)

\(\rm \frac{dx^{n}}{dx}= nx^{n-1}\)

Calculation:

We have to find the value of \(\rm \frac{d^2\tan x}{dx^2}\)

\(\rm \frac{d^2\tan x}{dx^2} = \frac{d}{dx} \left(\frac{d\tan x}{dx} \right )\)

\(\rm = \frac{d}{dx} \left(sec^2 x \right )\)

Apply chain rule, we get

\(\rm = \frac{d\sec^2 x}{d\sec x} × \frac{d\sec x}{dx}\)

= 2sec x . sec x tan x

= 2sec² x tan x