Solve c/c^2+3c+9-1/c-3+27/c^3-27

$$\frac { c } { c ^ { 2 } + 3 c + 9 } - \frac { 1 } { c - 3 } + \frac { 27 } { c ^ { 3 } - 27 }$$

Evaluate

$$-\frac{6}{c^{2}+3c+9}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of c^{2}+3c+9 and c-3 is \left(c-3\right)\left(c^{2}+3c+9\right). Multiply \frac{c}{c^{2}+3c+9} times \frac{c-3}{c-3}. Multiply \frac{1}{c-3} times \frac{c^{2}+3c+9}{c^{2}+3c+9}.

$$\frac{c\left(c-3\right)}{\left(c-3\right)\left(c^{2}+3c+9\right)}-\frac{c^{2}+3c+9}{\left(c-3\right)\left(c^{2}+3c+9\right)}+\frac{27}{c^{3}-27}$$

Since \frac{c\left(c-3\right)}{\left(c-3\right)\left(c^{2}+3c+9\right)} and \frac{c^{2}+3c+9}{\left(c-3\right)\left(c^{2}+3c+9\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{c\left(c-3\right)-\left(c^{2}+3c+9\right)}{\left(c-3\right)\left(c^{2}+3c+9\right)}+\frac{27}{c^{3}-27}$$

Do the multiplications in c\left(c-3\right)-\left(c^{2}+3c+9\right).

$$\frac{c^{2}-3c-c^{2}-3c-9}{\left(c-3\right)\left(c^{2}+3c+9\right)}+\frac{27}{c^{3}-27}$$

Combine like terms in c^{2}-3c-c^{2}-3c-9.

$$\frac{-6c-9}{\left(c-3\right)\left(c^{2}+3c+9\right)}+\frac{27}{c^{3}-27}$$

Factor c^{3}-27.

$$\frac{-6c-9}{\left(c-3\right)\left(c^{2}+3c+9\right)}+\frac{27}{\left(c-3\right)\left(c^{2}+3c+9\right)}$$

Since \frac{-6c-9}{\left(c-3\right)\left(c^{2}+3c+9\right)} and \frac{27}{\left(c-3\right)\left(c^{2}+3c+9\right)} have the same denominator, add them by adding their numerators.

$$\frac{-6c-9+27}{\left(c-3\right)\left(c^{2}+3c+9\right)}$$

Combine like terms in -6c-9+27.

$$\frac{-6c+18}{\left(c-3\right)\left(c^{2}+3c+9\right)}$$

Factor the expressions that are not already factored in \frac{-6c+18}{\left(c-3\right)\left(c^{2}+3c+9\right)}.

$$\frac{6\left(-c+3\right)}{\left(c-3\right)\left(c^{2}+3c+9\right)}$$

Extract the negative sign in 3-c.

$$\frac{-6\left(c-3\right)}{\left(c-3\right)\left(c^{2}+3c+9\right)}$$

Cancel out c-3 in both numerator and denominator.

$$\frac{-6}{c^{2}+3c+9}$$

Differentiate w.r.t. c

$$\frac{6\left(2c+3\right)}{\left(c^{2}+3c+9\right)^{2}}$$