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:
Apart from these remember the derivatives of some important functions like exponential, logarithmic, trigonometric etc..
Let’s solve now:
Given,
y=excos x
TO prove :
\(\frac{d^2y}{dx^2}=2e^xcos(x+\frac{\pi}{2})\)
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 = ex cos x
As, \(\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{dy}{dx})\)
So lets first find dy/dx and differentiate it again.
\(\therefore \frac{dy}{dx}=\frac{d}{dx}(e^xcosx)\)
Let u = ex and v = cos x
As, y = u*v
∴ Using product rule of differentiation:
Again differentiating w.r.t x:
Again using the product rule :