Question

Find the second order derivatives of the following functions: 

e^x sin 5x

Answer 1

 Basic Idea: Second order derivative is nothing but derivative of derivative i.e. \(\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{dy}{dx})\)

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:

 \(\frac{df}{dx}=\frac{dv}{dt}\times\frac{dt}{dx}\)

Product rule of differentiation- \(\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}\)

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

Let’s solve now:

Given, y = e^x sin 5x

We have to find \(\frac{d^2y}{dx^2}\) 

 As \(\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{dy}{dx})\) 

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

\(\therefore\frac{dy}{dx}=\frac{d}{dx}(log(sinx))\)

differentiating \(sin(logx)\) using cthe hain rule,

let, t = sin x and y = log t

Differentiating again with respect to x: