Solve 2(x+3/x-2)+3/(frac{x+3{x-2})-1}

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

Evaluate

$$x$$

Short Solution Steps

Express 2\times \frac{x+3}{x-2} as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{x-2}{x-2}.

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

Since \frac{2\left(x+3\right)}{x-2} and \frac{3\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2\left(x+3\right)+3\left(x-2\right).

$$\frac{\frac{2x+6+3x-6}{x-2}}{\frac{x+3}{x-2}-1}$$

Combine like terms in 2x+6+3x-6.

$$\frac{\frac{5x}{x-2}}{\frac{x+3}{x-2}-1}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-2}{x-2}.

$$\frac{\frac{5x}{x-2}}{\frac{x+3}{x-2}-\frac{x-2}{x-2}}$$

Since \frac{x+3}{x-2} and \frac{x-2}{x-2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{5x}{x-2}}{\frac{x+3-\left(x-2\right)}{x-2}}$$

Do the multiplications in x+3-\left(x-2\right).

$$\frac{\frac{5x}{x-2}}{\frac{x+3-x+2}{x-2}}$$

Combine like terms in x+3-x+2.

$$\frac{\frac{5x}{x-2}}{\frac{5}{x-2}}$$

Divide \frac{5x}{x-2} by \frac{5}{x-2} by multiplying \frac{5x}{x-2} by the reciprocal of \frac{5}{x-2}.

$$\frac{5x\left(x-2\right)}{\left(x-2\right)\times 5}$$

Cancel out 5\left(x-2\right) in both numerator and denominator.

$$x$$

Differentiate w.r.t. x

$$1$$

Steps Using Definition of a Derivative

Express 2\times \frac{x+3}{x-2} as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{x-2}{x-2}.

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

Since \frac{2\left(x+3\right)}{x-2} and \frac{3\left(x-2\right)}{x-2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2\left(x+3\right)+3\left(x-2\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{2x+6+3x-6}{x-2}}{\frac{x+3}{x-2}-1})$$

Combine like terms in 2x+6+3x-6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{5x}{x-2}}{\frac{x+3}{x-2}-1})$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-2}{x-2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{5x}{x-2}}{\frac{x+3}{x-2}-\frac{x-2}{x-2}})$$

Since \frac{x+3}{x-2} and \frac{x-2}{x-2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{5x}{x-2}}{\frac{x+3-\left(x-2\right)}{x-2}})$$

Do the multiplications in x+3-\left(x-2\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{5x}{x-2}}{\frac{x+3-x+2}{x-2}})$$

Combine like terms in x+3-x+2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{5x}{x-2}}{\frac{5}{x-2}})$$

Divide \frac{5x}{x-2} by \frac{5}{x-2} by multiplying \frac{5x}{x-2} by the reciprocal of \frac{5}{x-2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x\left(x-2\right)}{\left(x-2\right)\times 5})$$

Cancel out 5\left(x-2\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.

$$1$$