Question

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

Answer 1

Hint: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
 \[ \Rightarrow \dfrac{{dy}}{{dx}} = - 1.{(1 + {x^2})^{ - 1 - 1}}\dfrac{d}{{dx}}(1 + {x^2})\]
 \[ \Rightarrow \dfrac{{dy}}{{dx}} = - {(1 + {x^2})^{ - 2}}\left( {\dfrac{d}{{dx}}(1) +
\dfrac{d}{{dx}}({x^2})} \right)\]

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.