Solve 9/n^2-9n+1/9-n=

$$\frac { 9 } { n ^ { 2 } - 9 n } + \frac { 1 } { 9 - n } =$$

Evaluate

$$-\frac{1}{n}$$

Short Solution Steps

Factor n^{2}-9n.

$$\frac{9}{n\left(n-9\right)}+\frac{1}{9-n}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of n\left(n-9\right) and 9-n is n\left(n-9\right). Multiply \frac{1}{9-n} times \frac{-n}{-n}.

$$\frac{9}{n\left(n-9\right)}+\frac{-n}{n\left(n-9\right)}$$

Since \frac{9}{n\left(n-9\right)} and \frac{-n}{n\left(n-9\right)} have the same denominator, add them by adding their numerators.

$$\frac{9-n}{n\left(n-9\right)}$$

Extract the negative sign in 9-n.

$$\frac{-\left(n-9\right)}{n\left(n-9\right)}$$

Cancel out n-9 in both numerator and denominator.

$$\frac{-1}{n}$$

Differentiate w.r.t. n

$$\frac{1}{n^{2}}$$

Steps Using Definition of a Derivative

Factor n^{2}-9n.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{9}{n\left(n-9\right)}+\frac{1}{9-n})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of n\left(n-9\right) and 9-n is n\left(n-9\right). Multiply \frac{1}{9-n} times \frac{-n}{-n}.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{9}{n\left(n-9\right)}+\frac{-n}{n\left(n-9\right)})$$

Since \frac{9}{n\left(n-9\right)} and \frac{-n}{n\left(n-9\right)} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{9-n}{n\left(n-9\right)})$$

Extract the negative sign in 9-n.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{-\left(n-9\right)}{n\left(n-9\right)})$$

Cancel out n-9 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{-1}{n})$$

The derivative of ax^{n} is nax^{n-1}.

$$-\left(-1\right)n^{-1-1}$$

Multiply -1 times -1.

$$n^{-1-1}$$

Subtract 1 from -1.

$$n^{-2}$$