Question

How do you find the derivative of $\dfrac{1}{{(1 + {x^2})}}$

Answer 1

The derivative is the rate of change of the quantity at some point. Now here in this question we consider the given function as y and we differentiate the given function with respect to x. Hence, we can find the derivative of the function.

Complete step by step explanation:
Here in this question, we can find the derivative by two
methods.
Method 1: In this method consider the given function as y
$y = \dfrac{1}{{(1 + {x^2})}}$

The function which is in the denominator can be shifted to numerator, this function is rewritten as
$\Rightarrow y = {(1 + {x^2})^{ - 1}}$

Apply the differentiation to the function
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}{(1 + {x^2})^{ - 1}}$

We know that $\dfrac{d}{{dx}}({x^n}) = n.{x^{n - 1}}\dfrac{d}{{dx}}(x)$ , applying this
differentiation formula we have
$\Rightarrow \dfrac{{dy}}{{dx}} = - 1.{(1 + {x^2})^{ - 1 - 1}}\dfrac{d}{{dx}}(1 + {x^2})$

\dfrac{d}{{dx}}({x^2})} \right)$$ The differentiation of a constant function is zero and again we are considering the above differentiation formula.

\Rightarrow \dfrac{{dy}}{{dx}} = - {(1 + {x^2})^{ - 2}}\left( {0 + 2x\dfrac{d}{{dx}}(x)} \right)
\\
\Rightarrow \dfrac{{dy}}{{dx}} = - {(1 + {x^2})^{ - 2}}\left( {2x} \right) \\

This can be written in the form of fraction. $$ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{ - 2x}}{{{{(1 + {x^2})}^2}}}$$ Therefore, the derivative of $$\dfrac{1}{{(1 + {x^2})}}$$ is $$\dfrac{{ - 2x}}{{{{(1 + {x^2})}^2}}}$$ **Method 2:** In this method consider the given equation as y $$y = \dfrac{1}{{(1 + {x^2})}}$$ Now we will apply the quotient rule to the given function. The quotient rule is defined as $$\dfrac{d}{{dx}}\left( {\dfrac{u}{v}} \right) = \dfrac{{v\dfrac{{du}}{{dx}} - u\dfrac{{dv}}{{dx}}}}{{{v^2}}}$$ Here u is 1 and v is $$(1 + {x^2})$$ Applying the quotient rule we have $$ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{(1 + {x^2})\dfrac{d}{{dx}}(1) - (1)\dfrac{d}{{dx}}(1 + {x^2})}}{{{{(1 + {x^2})}^2}}}$$ The differentiation of a constant function is zero. $$\dfrac{d}{{dx}}({x^n}) = n.{x^{n - 1}}\dfrac{d}{{dx}}(x)$$ , applying this differentiation formula, we have $$ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{(1 + {x^2}).0 - (1)2x}}{{{{(1 + {x^2})}^2}}}$$ On simplification $$ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{ - 2x}}{{{{(1 + {x^2})}^2}}}$$ **Therefore, the derivative of $$\dfrac{1}{{(1 + {x^2})}}$$ is $$\dfrac{{ - 2x}}{{{{(1 + {x^2})}^2}}}$$** **Note:** To differentiate or to find the derivative of a function we use some standard differentiation formulas. The derivative is the rate of change of quantity, in this question we differentiate the given function with respect to x and find the derivative. The quotient rule is applied to solve this problem.