Second Derivative
Second Derivative
The second derivative of a function is the derivative of the first derivative and it indicates the change in gradient of the original function. The sign of the second derivative tells us if the gradient of the original function is increasing, decreasing or remaining constant.
To determine the second derivative of the function \(f(x)\), we differentiate \(f'(x)\) using the rules for differentiation.
\[f''(x) = \cfrac{d}{dx}[f'(x)]\]We also use the following notation for determining the second derivative of \(y\):
\[y'' = \cfrac{d}{dx}[\cfrac{dy}{dx}] = \cfrac{d^{2}y}{dx^{2}}\]
Example
Question
Calculate the second derivative for each of the following:
- \(k(x) = 2x^{3} - 4x^{2} + 9\)
- \(y = \cfrac{3}{x}\)
- \begin{align*} k'(x) &= 2(3x^{2}) - 4(2x) + 0 \\ &= 6x^{2} - 8x \\ & \\ k''(x) &= 6(2x) - 8 \\ &= 12x - 8 \end{align*}
- \begin{align*} y &= 3x^{-1} \\ & \\ \cfrac{dy}{dx} &= 3(-1x^{-2}) \\ &= -3x^{-2} \\ &= -\cfrac{3}{x^{2}} \\ & \\ \cfrac{d^{2}y}{dx^{2}} &= -3(-2x^{-3}) \\ &= \cfrac{6}{x^{3}} \end{align*}
This lesson is part of:
Differential Calculus
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