Question

What is the notation for the second derivative?

Answer 1

When a function is derived for the first time, it is called the first derivative. On deriving the first derivative again, we obtain the second derivative of the function. The derivative normally gives us the slope function at any point. The second derivative is called the derivative of the derivative.

Complete step by step solution:
Now let us learn more about the second derivative. The second derivative measures the instantaneous change of the first derivative. The sign of the second derivative signifies whether the slope of the tangent of the line to $f$ is increasing or decreasing. We can simply define the second derivative as the rate of change of the original function.
Now let us express the notation for the second derivative.
Let us consider a function $y=f\left( x \right)$.
As we know the notation of the first derivative is $y’=f\left( x \right)$.
Upon deriving this function again, we denote it as $y’’=f\left( x \right)$.
We can denote the notation in Leibniz notation form too as follows.
Let us consider the same function again.
Upon the first derivation, we denote it in the following way.

& y=f\left( x \right) \\\ & \Rightarrow \dfrac{dy}{dx}=f\left( x \right) \\\ \end{aligned}$$ Since, we need the notation for the second derive we denote it in the following way. $$\begin{aligned} & y=f\left( x \right) \\\ & \Rightarrow \dfrac{{{d}^{2}}y}{d{{x}^{2}}}=f\left( x \right) \\\ \end{aligned}$$ **Note:** If the first derivative of a point is zero, it is a local minimum or a local maximum. If the derivative of the same point is positive the point is a local minimum. If the derivative of the same point is negative, the point is local maximum.