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..
The idea of parametric form of differentiation:
If y = f (θ) and x = g(θ), i.e. y is a function of θ and x is also some other function of θ.
Then dy/dθ = f’(θ) and dx/dθ = g’(θ)
We notice a second order derivative in the expression to be proved so first take the step to find the second order derivative.
= a(-sin θ + θ cos θ + sin θ)
[ differentiated using product rule for θsinθ ]
= a θ cos θ ... eqn 4
Again differentiating w.r.t θ using product rule:-
Similarly,
Again differentiating w.r.t θ using product rule:-
Using equation 4 and 5 :
∴ again differentiating w.r.t x :-
Putting a value in the above equation-
We have :