Solve Page : ((3x)/(x - 1) * 1/(x + 1)) / (3/(x ^ 2 - 1)) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$( \frac { 3 x } { ( x - 1 ) } \times \frac { 1 } { ( x + 1 ) } ) \div \frac { 3 } { x ^ { 2 } - 1 }$$

Evaluate

$x$

Short Solution Steps

Multiply $\frac{3x}{x-1}$ times $\frac{1}{x+1}$ by multiplying numerator times numerator and denominator times denominator.

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

Divide $\frac{3x}{\left(x-1\right)\left(x+1\right)}$ by $\frac{3}{x^{2}-1}$ by multiplying $\frac{3x}{\left(x-1\right)\left(x+1\right)}$ by the reciprocal of $\frac{3}{x^{2}-1}$.

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

Cancel out $3$ in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out $\left(x-1\right)\left(x+1\right)$ in both numerator and denominator.

$$x$$

Differentiate w.r.t. x

$1$

Steps Using Definition of a Derivative

Multiply $\frac{3x}{x-1}$ times $\frac{1}{x+1}$ by multiplying numerator times numerator and denominator times denominator.

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

Divide $\frac{3x}{\left(x-1\right)\left(x+1\right)}$ by $\frac{3}{x^{2}-1}$ by multiplying $\frac{3x}{\left(x-1\right)\left(x+1\right)}$ by the reciprocal of $\frac{3}{x^{2}-1}$.

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

Cancel out $3$ in both numerator and denominator.

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

Factor the expressions that are not already factored in $\frac{x\left(x^{2}-1\right)}{\left(x-1\right)\left(x+1\right)}$.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)})$$

Cancel out $\left(x-1\right)\left(x+1\right)$ in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x)$$

The derivative of $ax^{n}$ is $nax^{n-1}$.

$$x^{1-1}$$

Subtract $1$ from $1$.

$$x^{0}$$

For any term $t$ except $0$, $t^{0}=1$.

$$1$$