Find the second order derivatives of each of the following functions:

e^6x cos 3x

√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^6x cos 3x

We have to find

As,

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

∴

Let u = e^6x and v = cos 3x

As, y = uv

∴ Using product rule of differentiation:

∴

[ ∵ ]

Again differentiating w.r.t x:

Again using the product rule :