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Q.
If $y=f(x)$, then the second order derive of $y$ w.r.t $x$ is denoted by
Continuity and Differentiability
Solution:
Let $y=f(x)$
Then, $ \frac{d y}{d x}=f^{\prime}(x) .....$(i)
If $f^{\prime}(x)$ is differentiable, we may differentiate Eq. (i) again w.r.t.x. Then, the left hand side becomes $\frac{d}{d x}\left(\frac{d y}{d x}\right)$ which is called the second order derivative of $y$ w.r.t. $x$ and is denoted by $\frac{d^2 y}{d x^2}$.
The second order derivative of $f(x)$ is denoted by $f^{\prime \prime}(x)$. It is also denoted by. $D^2 y$ or $y^{\prime \prime}$ or $y_2$, if $y=f(x)$. We remark that higher order derivatives may be defined similarly.