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Quotient Rule of Differentiation Calculator

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 Difficult Problems

1

Here, we show you a step-by-step solved example of quotient rule of differentiation. This solution was automatically generated by our smart calculator:

$\frac{d}{dx}\left(\frac{x}{x^2+1}\right)$
2

Apply the quotient rule for differentiation, which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{\frac{d}{dx}\left(x\right)\left(x^2+1\right)-x\frac{d}{dx}\left(x^2+1\right)}{\left(x^2+1\right)^2}$
3

The derivative of the linear function is equal to $1$

$\frac{x^2+1-x\frac{d}{dx}\left(x^2+1\right)}{\left(x^2+1\right)^2}$
4

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{x^2+1-x\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(1\right)\right)}{\left(x^2+1\right)^2}$
5

Simplify the product $-(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(1\right))$

$\frac{x^2+1+\left(-\frac{d}{dx}\left(x^2\right)-\frac{d}{dx}\left(1\right)\right)x}{\left(x^2+1\right)^2}$
6

The derivative of the constant function ($1$) is equal to zero

$\frac{x^2+1+\left(-\frac{d}{dx}\left(x^2\right)+0\right)x}{\left(x^2+1\right)^2}$
7

$x+0=x$, where $x$ is any expression

$\frac{x^2+1-\frac{d}{dx}\left(x^2\right)x}{\left(x^2+1\right)^2}$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$-2x^{\left(2-1\right)}x$

Subtract the values $2$ and $-1$

$-2x\cdot x$
8

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\frac{x^2+1- 2x\cdot x}{\left(x^2+1\right)^2}$

Multiply $-1$ times $2$

$\frac{x^2+1-2x\cdot x}{\left(x^2+1\right)^2}$

When multiplying two powers that have the same base ($x$), you can add the exponents

$\frac{x^2+1-2x^2}{\left(x^2+1\right)^2}$
9

Simplify the derivative

$\frac{x^2+1-2x^2}{\left(x^2+1\right)^2}$

 Final answer to the problem

$\frac{x^2+1-2x^2}{\left(x^2+1\right)^2}$

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