Solve x+h/x+h+1-x/x+1/h

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+h+1 and x+1 is \left(x+1\right)\left(x+h+1\right). Multiply \frac{x+h}{x+h+1} times \frac{x+1}{x+1}. Multiply \frac{x}{x+1} times \frac{x+h+1}{x+h+1}.

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

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

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

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

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

Combine like terms in x^{2}+x+hx+h-x^{2}-xh-x.

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

Express \frac{\frac{h}{\left(x+1\right)\left(x+h+1\right)}}{h} as a single fraction.

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

Cancel out h in both numerator and denominator.

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

Apply the distributive property by multiplying each term of x+1 by each term of x+h+1.

$$\frac{1}{x^{2}+xh+x+x+h+1}$$

Combine x and x to get 2x.

$$\frac{1}{x^{2}+xh+2x+h+1}$$

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+h+1 and x+1 is \left(x+1\right)\left(x+h+1\right). Multiply \frac{x+h}{x+h+1} times \frac{x+1}{x+1}. Multiply \frac{x}{x+1} times \frac{x+h+1}{x+h+1}.

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

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

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

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

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

Combine like terms in x^{2}+x+hx+h-x^{2}-xh-x.

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

Express \frac{\frac{h}{\left(x+1\right)\left(x+h+1\right)}}{h} as a single fraction.

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

Cancel out h in both numerator and denominator.

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

Apply the distributive property by multiplying each term of x+1 by each term of x+h+1.

$$\frac{1}{x^{2}+xh+x+x+h+1}$$

Combine x and x to get 2x.

$$\frac{1}{x^{2}+xh+2x+h+1}$$