Answer :

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


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