Find the second order derivatives of each of the following functions:
ex sin 5x
√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 = ex sin 5x
We have to find
As,
So lets first find dy/dx and differentiate it again.
∴
Let u = ex and v = sin 5x
As, y = uv
∴ Using product rule of differentiation:
∴
[ ∵ ]
Again differentiating w.r.t x:
Again using the product rule :
[∵