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

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

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 1+x+h and 1+x is \left(x+1\right)\left(x+h+1\right). Multiply \frac{1}{1+x+h} times \frac{x+1}{x+1}. Multiply \frac{1}{1+x} times \frac{x+h+1}{x+h+1}.

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

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

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

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

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

Combine like terms in x+1-x-h-1.

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

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 1+x+h and 1+x is \left(x+1\right)\left(x+h+1\right). Multiply \frac{1}{1+x+h} times \frac{x+1}{x+1}. Multiply \frac{1}{1+x} times \frac{x+h+1}{x+h+1}.

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

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

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

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

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

Combine like terms in x+1-x-h-1.

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

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.

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