If y = e^x cosx, prove that   (CBSE 2012)

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=e^x cos x

TO prove :

Clearly from the expression to be proved we can easily observe that we need to just find the second derivative of given function.

Given, y = e^x cos x

We have to find

As

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

∴

Let u = e^x and v = cos x

As, y = u*v

∴ Using product rule of differentiation:

∴

[ ∵ ]

Again differentiating w.r.t x:

Again using the product rule :

[∵

[∵ –sin x = cos (x + π/2)]