Question

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

Answer 1

In the given equation it shows the operation of the division. We can say that to use the quotient rule we have to multiply the denominator with the derivative of the numerator and also subtract the product of the numerator and also with its derivative of the denominator. After that divide the whole thing by square of denominator. In this we have to take only two derivatives. Also you have to apply chain rule which will allow you to find derivatives of any function.

Complete step-by-step answer:
As we have,
$\Rightarrow$ $\dfrac{x}{\sqrt{{{x}^{2}}+1}}$
Hereby given the equation we see that it is under the operation of division.
So, we have to apply quotient rule we have,
By applying quotient rule,
$\Rightarrow$ $\dfrac{d}{dx}\left[ \dfrac{f(x)}{g(x)} \right]$
$\Rightarrow$ $g(x).\dfrac{d}{dx}f(x)-f(x).\dfrac{d}{dx}g(x){{\left[ g\left( x \right) \right]}^{2}}$
Now apply this rule in the given equation we get,
$\Rightarrow$ $\dfrac{d}{dx}\left( \dfrac{x}{\sqrt{{{x}^{2}}+1}} \right)$
$\Rightarrow$ $\sqrt{{{x}^{2}}+1}\dfrac{\dfrac{d}{dx}\left( x \right)-x.\dfrac{d}{dx}\left( \sqrt{{{x}^{2}}+1} \right)}{{{\left( \sqrt{{{x}^{2}}+1} \right)}^{2}}}...(i)$
After this we use the chain rule to calculated we get,
$\Rightarrow$ $\dfrac{d}{dx}\left( \sqrt{{{x}^{2}}+1} \right)=2x.\dfrac{1}{2\sqrt{{{x}^{2}}+1}}=\dfrac{x}{\sqrt{{{x}^{2}}+1}}...(ii)$
Now, we have to substitute the equation, $(ii)$ in $(i)$ we get,
$\Rightarrow$ $\dfrac{d}{dx}\left( \dfrac{x}{\sqrt{{{x}^{2}}+1}} \right)=\dfrac{\sqrt{{{x}^{2}}+1}-x\dfrac{d}{dx}\left( \sqrt{{{x}^{2}}+1} \right)}{{{x}^{2}}+1}$
$\Rightarrow$ $\dfrac{\sqrt{{{x}^{2}}+1}-x.\dfrac{x}{\sqrt{{{x}^{2}}+1}}}{{{x}^{2}}+1}$
$\left( \dfrac{d}{dx}\left( x \right)=x \right)$
$\Rightarrow$ $\dfrac{d}{dx}\left( \dfrac{x}{\sqrt{{{x}^{2}}+1}} \right)=\dfrac{{{x}^{2}}+1-{{x}^{2}}}{\left( {{x}^{2}}+1 \right)\sqrt{{{x}^{2}}+1}}$
$\Rightarrow$ $\dfrac{1}{\left( {{x}^{2}}+1 \right)\sqrt{{{x}^{2}}+1}}$
$\Rightarrow$ $\dfrac{d}{dx}\left( \dfrac{x}{\sqrt{{{x}^{2}}+1}} \right)=\dfrac{1}{{{\left( {{x}^{2}}+1 \right)}^{\dfrac{3}{2}}}}$

Hence, the derivative of $\dfrac{x}{\sqrt{{{x}^{2}}+1}}$ is $\dfrac{1}{{{\left( {{x}^{2}}+1 \right)}^{\dfrac{3}{2}}}}$

Additional Information:
The quotient rule we have applies to extend the power rule for negative integer exponents or we can say that it is for finding the derivative of a quotient of function. In the formula we see that the derivative of the quotient of a function is the denominator function which is multiplied by denominator. At last all the terms are divided by the square of denominator.
$\Rightarrow$ $\dfrac{d}{dx}\left[ \dfrac{f(x)}{g(x)} \right]=\dfrac{g(x).\dfrac{d}{dx}f(x)-f(x).\dfrac{d}{dx}g(x)}{{{\left[ g\left( x \right) \right]}^{2}}}$
Where, as $g(x)$ is the denominator and $f(x)$ is the numerator of function.
And in chain rule it allows us to find the derivative or any function. It is one of the most useful rules in differential problems.

Note:
When you have any problem for solving the first check it is in which form due to that we get the idea for applying any rule in the question. Similarly here we use the quotient rule, so check the formula you use. Also we have to use the derivatives in the formula after solving the problem and analyse all the possibilities where we have got wrong. In the case of chain rule the function can be expressed as composition of a function that is $f(x)$ and $g(x)$ such as $f\left( g\left( x \right) \right)$. These are the same important steps we have to know while solving these problems.