If y = x + tan x, show that: cos²

Basic idea:

√Second order derivative is nothing but derivative of derivative i.e.

√The idea of chain rule of differentiation: If f is any real-valued function which is the composition of two functions u and v, i.e. f = v(u(x)). For the sake of simplicity just assume t = u(x)

Then f = v(t). By chain rule, we can write the derivative of f w.r.t to x as:

√Product rule of differentiation-

√Apart from these remember the derivatives of some important functions like exponential, logarithmic, trigonometric etc..

Let’s solve now:

Given, y = x+ tan x …..equation 1

As we have to prove: cos²

We notice a second-order derivative in the expression to be proved so first take the step to find the second order derivative.

Let’s find

As

So lets first find dy/dx and differentiate it again.

∴ [∵ tan x) = sec² x & ]

∴

Differentiating again with respect to x :

[ differentiated sec²x using chain rule, let t = sec x and z = t² ∴ ]

……….equation 2

As we got an expression for the second order, as we need cos²x term with

Multiply both sides of equation 1 with cos²x:

∴ we have,

[∵ cos x × sec x = 1]

From equation 1:

tan x = y – x

….proved