Question

If $y=f(x)$, then the second order derive of $y$ w.r.t $x$ is denoted by

• A. $\frac{d^{2}y}{d{x}^2}$ or $y^{\prime \prime }$
• B. $f^{\prime }(x)$
• C. $y_1$
• D. None of these

Answer 1

Answer: (A)

Let $y=f(x)$
Then, $\frac{dy}{dx}=f^{\prime }(x).....$(i)
If $f^{\prime }(x)$ is differentiable, we may differentiate Eq. (i) again w.r.t.x. Then, the left hand side becomes $\frac{d}{dx}\left(\frac{dy}{dx}\right)$ which is called the second order derivative of $y$ w.r.t. $x$ and is denoted by $\frac{d^{2}y}{d{x}^2}$.
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.