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 D2y or y'' or y2 if y =f(x).

View the Topic in this video From 06:00 To 10:26

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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 = eax + b \Rightarrow \frac{d^{n}y}{dx^{n}}=a^{n} \cdot e^{(ax+b)}