Solve 1/x+6/1+frac{1{x+6}}

$$\frac { \frac { 1 } { x + 6 } } { 1 + \frac { 1 } { x + 6 } }$$

Evaluate

$$\frac{1}{x+7}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x+6}{x+6}.

$$\frac{\frac{1}{x+6}}{\frac{x+6}{x+6}+\frac{1}{x+6}}$$

Since \frac{x+6}{x+6} and \frac{1}{x+6} have the same denominator, add them by adding their numerators.

$$\frac{\frac{1}{x+6}}{\frac{x+6+1}{x+6}}$$

Combine like terms in x+6+1.

$$\frac{\frac{1}{x+6}}{\frac{x+7}{x+6}}$$

Divide \frac{1}{x+6} by \frac{x+7}{x+6} by multiplying \frac{1}{x+6} by the reciprocal of \frac{x+7}{x+6}.

$$\frac{x+6}{\left(x+6\right)\left(x+7\right)}$$

Cancel out x+6 in both numerator and denominator.

$$\frac{1}{x+7}$$

Differentiate w.r.t. x

$$-\frac{1}{\left(x+7\right)^{2}}$$

Steps Using Chain Rule

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x+6}{x+6}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{x+6}}{\frac{x+6}{x+6}+\frac{1}{x+6}})$$

Since \frac{x+6}{x+6} and \frac{1}{x+6} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{x+6}}{\frac{x+6+1}{x+6}})$$

Combine like terms in x+6+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{x+6}}{\frac{x+7}{x+6}})$$

Divide \frac{1}{x+6} by \frac{x+7}{x+6} by multiplying \frac{1}{x+6} by the reciprocal of \frac{x+7}{x+6}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x+6}{\left(x+6\right)\left(x+7\right)})$$

Cancel out x+6 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x+7})$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\left(x^{1}+7\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{1}+7)$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-\left(x^{1}+7\right)^{-2}x^{1-1}$$

Simplify.

$$-x^{0}\left(x^{1}+7\right)^{-2}$$

For any term t, t^{1}=t.

$$-x^{0}\left(x+7\right)^{-2}$$

For any term t except 0, t^{0}=1.

$$-\left(x+7\right)^{-2}$$