The derivative of sec² x with respect to tan² x is

Question

Question

Answer 1

Concept:

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

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

Steps for derivatives of functions expressed in the parametric form:

First of all, we write the given functions u and v in terms of the parameter x.
Using differentiation find out du/dx and dv/dx.
Then by using the formula used for solving functions in parametric form i.e.
Lastly substituting the values of du/dx and dv/dx and simplify to obtain the result.

Calculation:

Let u = sec² x and v = tan² x

Differentiating with respect to x, we get

\(\rm \Rightarrow \frac{du}{dx} = 2\rm\sec x \times \sec x \tan x \;and\; \frac{dv}{dx} = 2\rm \tan x \times \sec^2 x\)

Now,

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

\(= \rm \frac{2\rm\sec x \times \sec x \tan x}{ 2\rm \tan x \times \sec^2 x }=1\)