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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x-x-h.

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

Multiply \frac{1}{h} times \frac{-h}{x\left(x+h\right)} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out h in both numerator and denominator.

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

Use the distributive property to multiply x by x+h.

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x-x-h.

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

Multiply \frac{1}{h} times \frac{-h}{x\left(x+h\right)} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out h in both numerator and denominator.

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

Use the distributive property to multiply x by x+h.

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