Solve 3x/x-1/frac{3x{x-1}-3}

$$\frac { \frac { 3 x } { x - 1 } } { \frac { 3 x } { x - 1 } - 3 }$$

Evaluate

$$x$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{x-1}{x-1}.

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

Since \frac{3x}{x-1} and \frac{3\left(x-1\right)}{x-1} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3x-3\left(x-1\right).

$$\frac{\frac{3x}{x-1}}{\frac{3x-3x+3}{x-1}}$$

Combine like terms in 3x-3x+3.

$$\frac{\frac{3x}{x-1}}{\frac{3}{x-1}}$$

Divide \frac{3x}{x-1} by \frac{3}{x-1} by multiplying \frac{3x}{x-1} by the reciprocal of \frac{3}{x-1}.

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

Cancel out 3\left(x-1\right) in both numerator and denominator.

$$x$$

Differentiate w.r.t. x

$$1$$

Steps Using Definition of a Derivative

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{x-1}{x-1}.

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

Since \frac{3x}{x-1} and \frac{3\left(x-1\right)}{x-1} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3x-3\left(x-1\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{3x}{x-1}}{\frac{3x-3x+3}{x-1}})$$

Combine like terms in 3x-3x+3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{3x}{x-1}}{\frac{3}{x-1}})$$

Divide \frac{3x}{x-1} by \frac{3}{x-1} by multiplying \frac{3x}{x-1} by the reciprocal of \frac{3}{x-1}.

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

Cancel out 3\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.

$$1$$