## Continuity and Differentiability

# Second Order Derivative

If f'(*x*) is differentiable, w.r.t. *x*. Then, the left hand side becomes \frac{d}{dx}\left(\frac{dy}{dx}\right) which is called the *second order derivative* of *y* w.r.t. *x* and is denoted by \frac{d^2y}{dx^2}. The second order derivative of f(*x*) is denoted by f''(*x*). It is also denoted by D^{2}y or y'' or y_{2} if y =f(x).

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- If y=\frac{f(x)}{1+\frac{f'(x)}{1+y}}, then \frac{dy}{dx}=\frac{(1+y)f'(x)-y f^{"}(x)}{(1+2y)-f(x)+f^{"}(x)}
- If y = sin (ax + b) \Rightarrow \frac{d^{n}y}{dx^{n}}=\sin \left(\frac{n\pi}{2}+(ax+b)\right)
- y = cos (ax + b) \Rightarrow \frac{d^{n}y}{dx^{n}}=\cos \left(\frac{n\pi}{2}+(ax+b)\right)
- y = e
^{ax + b}\Rightarrow \frac{d^{n}y}{dx^{n}}=a^{n} \cdot e^{(ax+b)}