Question:
Find the second order derivatives of the function
\(tan^{-1}x\)
Find the second order derivatives of the function
\(tan^{-1}x\)
\(tan^{-1}x\)
Updated On: Sep 12, 2023
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Solution and Explanation
The correct answer is \(=\frac{-2x}{(1+x^2)^2}\)
Let \(y=tan^{-1}x\)
Then,
\(\frac{dy}{dx}=\frac{d}{dx}(tan^{-1}x)=\frac{1}{1+x^2}\)
\(∴\frac{d^2y}{dx^2}=\frac{d}{dx}[\frac{1}{1+x^2}]\)
\(=\frac{d}{dx}(1+x^2)^{-1}\)
\(=(-1).(1+x^2)^{-2}.\frac{d}{dx}(1+x^2)\)
\(=\frac{-1}{(1+x^2)^2}\times 2x\)
\(=\frac{-2x}{(1+x^2)^2}\)
Let \(y=tan^{-1}x\)
Then,
\(\frac{dy}{dx}=\frac{d}{dx}(tan^{-1}x)=\frac{1}{1+x^2}\)
\(∴\frac{d^2y}{dx^2}=\frac{d}{dx}[\frac{1}{1+x^2}]\)
\(=\frac{d}{dx}(1+x^2)^{-1}\)
\(=(-1).(1+x^2)^{-2}.\frac{d}{dx}(1+x^2)\)
\(=\frac{-1}{(1+x^2)^2}\times 2x\)
\(=\frac{-2x}{(1+x^2)^2}\)
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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 \(f’’(x)\text{ or }\frac{d^2y}{dx^2}\).