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

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

Evaluate

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

Short Solution Steps

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

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

Since \frac{xx}{x} and \frac{1}{x} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in xx-1.

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

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

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

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

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

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

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

Combine like terms in x^{2}+xh+xh+h^{2}-1.

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

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{x^{2}+h^{2}+2xh-1}{x+h} times \frac{x}{x}. Multiply \frac{x^{2}-1}{x} times \frac{x+h}{x+h}.

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

Since \frac{\left(x^{2}+h^{2}+2xh-1\right)x}{x\left(x+h\right)} and \frac{\left(x^{2}-1\right)\left(x+h\right)}{x\left(x+h\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x^{2}+h^{2}+2xh-1\right)x-\left(x^{2}-1\right)\left(x+h\right).

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

Combine like terms in x^{3}+h^{2}x+2x^{2}h-x-x^{3}-x^{2}h+x+h.

$$\frac{\frac{h+x^{2}h+h^{2}x}{x\left(x+h\right)}}{h}$$

Express \frac{\frac{h+x^{2}h+h^{2}x}{x\left(x+h\right)}}{h} as a single fraction.

$$\frac{h+x^{2}h+h^{2}x}{x\left(x+h\right)h}$$

Factor the expressions that are not already factored.

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

Cancel out h in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{2}+hx+1}{x^{2}+hx}$$

Expand

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

Short Solution Steps

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

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

Since \frac{xx}{x} and \frac{1}{x} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in xx-1.

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

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

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

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

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

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

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

Combine like terms in x^{2}+xh+xh+h^{2}-1.

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

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{x^{2}+h^{2}+2xh-1}{x+h} times \frac{x}{x}. Multiply \frac{x^{2}-1}{x} times \frac{x+h}{x+h}.

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

Since \frac{\left(x^{2}+h^{2}+2xh-1\right)x}{x\left(x+h\right)} and \frac{\left(x^{2}-1\right)\left(x+h\right)}{x\left(x+h\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x^{2}+h^{2}+2xh-1\right)x-\left(x^{2}-1\right)\left(x+h\right).

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

Combine like terms in x^{3}+h^{2}x+2x^{2}h-x-x^{3}-x^{2}h+x+h.

$$\frac{\frac{h+x^{2}h+h^{2}x}{x\left(x+h\right)}}{h}$$

Express \frac{\frac{h+x^{2}h+h^{2}x}{x\left(x+h\right)}}{h} as a single fraction.

$$\frac{h+x^{2}h+h^{2}x}{x\left(x+h\right)h}$$

Factor the expressions that are not already factored.

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

Cancel out h in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{2}+hx+1}{x^{2}+hx}$$