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Derivative of 1x2+1\frac{1}{x^{2} + 1}

The calculator will find the derivative of 1x2+1\frac{1}{x^{2} + 1}, with steps shown.

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Your Input

Find ddx(1x2+1)\frac{d}{dx} \left(\frac{1}{x^{2} + 1}\right).

Solution

The function 1x2+1\frac{1}{x^{2} + 1} is the composition f(g(x))f{\left(g{\left(x \right)} \right)} of two functions f(u)=1uf{\left(u \right)} = \frac{1}{u} and g(x)=x2+1g{\left(x \right)} = x^{2} + 1.

Apply the chain rule ddx(f(g(x)))=ddu(f(u))ddx(g(x))\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right):

(ddx(1x2+1))=(ddu(1u)ddx(x2+1)){\color{red}\left(\frac{d}{dx} \left(\frac{1}{x^{2} + 1}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\frac{1}{u}\right) \frac{d}{dx} \left(x^{2} + 1\right)\right)}

Apply the power rule ddu(un)=nun−1\frac{d}{du} \left(u^{n}\right) = n u^{n - 1} with n=−1n = -1:

(ddu(1u))ddx(x2+1)=(−1u2)ddx(x2+1){\color{red}\left(\frac{d}{du} \left(\frac{1}{u}\right)\right)} \frac{d}{dx} \left(x^{2} + 1\right) = {\color{red}\left(- \frac{1}{u^{2}}\right)} \frac{d}{dx} \left(x^{2} + 1\right)

Return to the old variable:

−ddx(x2+1)(u)2=−ddx(x2+1)(x2+1)2- \frac{\frac{d}{dx} \left(x^{2} + 1\right)}{{\color{red}\left(u\right)}^{2}} = - \frac{\frac{d}{dx} \left(x^{2} + 1\right)}{{\color{red}\left(x^{2} + 1\right)}^{2}}

The derivative of a sum/difference is the sum/difference of derivatives:

−(ddx(x2+1))(x2+1)2=−(ddx(x2)+ddx(1))(x2+1)2- \frac{{\color{red}\left(\frac{d}{dx} \left(x^{2} + 1\right)\right)}}{\left(x^{2} + 1\right)^{2}} = - \frac{{\color{red}\left(\frac{d}{dx} \left(x^{2}\right) + \frac{d}{dx} \left(1\right)\right)}}{\left(x^{2} + 1\right)^{2}}

Apply the power rule ddx(xn)=nxn−1\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1} with n=2n = 2:

−(ddx(x2))+ddx(1)(x2+1)2=−(2x)+ddx(1)(x2+1)2- \frac{{\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} + \frac{d}{dx} \left(1\right)}{\left(x^{2} + 1\right)^{2}} = - \frac{{\color{red}\left(2 x\right)} + \frac{d}{dx} \left(1\right)}{\left(x^{2} + 1\right)^{2}}

The derivative of a constant is 00:

−2x+(ddx(1))(x2+1)2=−2x+(0)(x2+1)2- \frac{2 x + {\color{red}\left(\frac{d}{dx} \left(1\right)\right)}}{\left(x^{2} + 1\right)^{2}} = - \frac{2 x + {\color{red}\left(0\right)}}{\left(x^{2} + 1\right)^{2}}

Thus, ddx(1x2+1)=−2x(x2+1)2\frac{d}{dx} \left(\frac{1}{x^{2} + 1}\right) = - \frac{2 x}{\left(x^{2} + 1\right)^{2}}.

Answer

ddx(1x2+1)=−2x(x2+1)2\frac{d}{dx} \left(\frac{1}{x^{2} + 1}\right) = - \frac{2 x}{\left(x^{2} + 1\right)^{2}}A

Related Notes: Definition of Derivative , Derivatives of Elementary Functions , Table of Derivatives , Related Rates , Studying Derivative Graphically , Optimization Problems , Applications to Economics , Constant Multiple Rule , Sum and Difference Rules , Product Rule , Quotient Rule , Chain Rule , Derivative of Inverse Function