Solve 1/x+h-1/1+xquad70.frac{1/sqrt{x+h}-1/sqrt{x}}{h}

$$\frac { 1 } { x + h } - \frac { 1 } { 1 + x } \quad 70 . \frac { \frac { 1 } { \sqrt { x + h } } - \frac { 1 } { \sqrt { x } } } { h }$$

Evaluate

$$\frac{1-70h-69x}{\left(x+1\right)\left(x+h\right)}$$

Short Solution Steps

Express \frac{1}{1+x}\times 70 as a single fraction.

$$\frac{1}{x+h}-\frac{70}{1+x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+h and 1+x is \left(x+1\right)\left(x+h\right). Multiply \frac{1}{x+h} times \frac{x+1}{x+1}. Multiply \frac{70}{1+x} times \frac{x+h}{x+h}.

$$\frac{x+1}{\left(x+1\right)\left(x+h\right)}-\frac{70\left(x+h\right)}{\left(x+1\right)\left(x+h\right)}$$

Since \frac{x+1}{\left(x+1\right)\left(x+h\right)} and \frac{70\left(x+h\right)}{\left(x+1\right)\left(x+h\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x+1-70\left(x+h\right)}{\left(x+1\right)\left(x+h\right)}$$

Do the multiplications in x+1-70\left(x+h\right).

$$\frac{x+1-70x-70h}{\left(x+1\right)\left(x+h\right)}$$

Combine like terms in x+1-70x-70h.

$$\frac{-69x+1-70h}{\left(x+1\right)\left(x+h\right)}$$

Expand \left(x+1\right)\left(x+h\right).

$$\frac{-69x+1-70h}{x^{2}+hx+x+h}$$

Differentiate w.r.t. x

$$\frac{69x^{2}+140hx-2x+70h^{2}-1}{\left(\left(x+1\right)\left(x+h\right)\right)^{2}}$$