Find the derivative of x/(x^2-1)

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

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Answer

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

Step by step solution

Problem

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

Applying the quotient rule 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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

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

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

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

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

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

The derivative of the constant function is equal to zero

$\frac{1\left(x^2-1\right)-x\left(0+\frac{d}{dx}\left(x^2\right)\right)}{\left(x^2-1\right)^2}$
5

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

$\frac{1\left(x^2-1\right)-x\left(0+2x\right)}{\left(x^2-1\right)^2}$
6

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

$\frac{1\left(x^2-1\right)-1\cdot 2x\cdot x}{\left(x^2-1\right)^2}$
7

Multiply $2$ times $-1$

$\frac{1\left(x^2-1\right)-2x\cdot x}{\left(x^2-1\right)^2}$
8

Any expression multiplied by $1$ is equal to itself

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

When multiplying exponents with same base you can add the exponents

$\frac{-2x^2-1+x^2}{\left(x^2-1\right)^2}$
10

Adding $-2x^2$ and $x^2$

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

Answer

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

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Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.29 seconds

Views:

86