Solve 3x/x-4/frac{5{x-4}-6/x-4}=

$$\frac { \frac { 3 x } { x - 4 } } { \frac { 5 } { x - 4 } - \frac { 6 } { x - 4 } } =$$

Evaluate

$$-3x$$

Solution Steps

Since \frac{5}{x-4} and \frac{6}{x-4} have the same denominator, subtract them by subtracting their numerators. Subtract 6 from 5 to get -1.

$$\frac{\frac{3x}{x-4}}{\frac{-1}{x-4}}$$

Divide \frac{3x}{x-4} by \frac{-1}{x-4} by multiplying \frac{3x}{x-4} by the reciprocal of \frac{-1}{x-4}.

$$\frac{3x\left(x-4\right)}{\left(x-4\right)\left(-1\right)}$$

Cancel out x-4 in both numerator and denominator.

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

Anything divided by -1 gives its opposite.

$$-3x$$

Differentiate w.r.t. x

$$-3$$

Steps Using Definition of a Derivative

Since \frac{5}{x-4} and \frac{6}{x-4} have the same denominator, subtract them by subtracting their numerators. Subtract 6 from 5 to get -1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{3x}{x-4}}{\frac{-1}{x-4}})$$

Divide \frac{3x}{x-4} by \frac{-1}{x-4} by multiplying \frac{3x}{x-4} by the reciprocal of \frac{-1}{x-4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3x\left(x-4\right)}{\left(x-4\right)\left(-1\right)})$$

Cancel out x-4 in both numerator and denominator.

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

Anything divided by -1 gives its opposite.

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

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

$$-3x^{1-1}$$

Subtract 1 from 1.

$$-3x^{0}$$

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

$$-3$$