Find the second order derivatives of the function
\(x^{20}\)
\(x^{20}\)
Solution and Explanation
Let \(y=x^{20}\)
Then,
\(\frac{dy}{dx}=\frac{d}{dx}(x^{20})=20x^{19}\)
\(∴\frac{d^2y}{dx^2}=\frac{d}{dx}(20x^{19})=20\frac{d}{dx}(x^{19})=20.19.x^{18}=380x^{18}\)
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Concepts Used:
Second-Order Derivative
The Second-Order Derivative is the derivative of the first-order derivative of the stated (given) function. For instance, acceleration is the second-order derivative of the distance covered with regard to time and tells us the rate of change of velocity.
As well as the first-order derivative tells us about the slope of the tangent line to the graph of the given function, the second-order derivative explains the shape of the graph and its concavity.
The second-order derivative is shown using \(f’’(x)\text{ or }\frac{d^2y}{dx^2}\).