Question

Explain the procedure to find $\frac{d{}^{2}y}{d{x}^2}$ [second order derivative] of any function y=f(x).

Answer 1

Definition

• Double derivative or $\frac{d{}^{2}y}{d{x}^2}$ is defined as derivative of $\frac{dy}{dx}$ .

Example

Let the function be $f\left(x\right)=y=x^2+x$ .

• Now, the first derivative is -

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=\frac{d(x^2+x)}{dx} \\ \Rightarrow \frac{dy}{dx}=2x+1 \end{gathered}$$ (differentiating y with respect to x)

• Now, second derivative is -

$$\begin{gathered} \Rightarrow \frac{d{}^{2}y}{d{x}^2}=\frac{d(2x+1)}{dx} \\ \Rightarrow \frac{d{}^{2}y}{d{x}^2}=2 \end{gathered}$$ (differentiating $\frac{dy}{dx}$ with respect to x)

Hence $\frac{d{}^{2}y}{d{x}^2}$ of f(x) is 2 .