Solve (1/a+h+1)-(1/a+1)/h

$$\frac { ( \frac { 1 } { a + h + 1 } ) - ( \frac { 1 } { a + 1 } ) } { h }$$

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Cancel out h in both numerator and denominator.

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

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

$$\frac{-1}{a^{2}+ah+a+a+h+1}$$

Combine a and a to get 2a.

$$\frac{-1}{a^{2}+ah+2a+h+1}$$

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Cancel out h in both numerator and denominator.

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

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

$$\frac{-1}{a^{2}+ah+a+a+h+1}$$

Combine a and a to get 2a.

$$\frac{-1}{a^{2}+ah+2a+h+1}$$