Solve 3/3m+4/2m+12

$$\frac { 3 } { 3 m } + \frac { 4 } { 2 m + 12 }$$

Evaluate

$$\frac{3\left(m+2\right)}{m\left(m+6\right)}$$

Short Solution Steps

Factor 2m+12.

$$\frac{3}{3m}+\frac{4}{2\left(m+6\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3m and 2\left(m+6\right) is 6m\left(m+6\right). Multiply \frac{3}{3m} times \frac{2\left(m+6\right)}{2\left(m+6\right)}. Multiply \frac{4}{2\left(m+6\right)} times \frac{3m}{3m}.

$$\frac{3\times 2\left(m+6\right)}{6m\left(m+6\right)}+\frac{4\times 3m}{6m\left(m+6\right)}$$

Since \frac{3\times 2\left(m+6\right)}{6m\left(m+6\right)} and \frac{4\times 3m}{6m\left(m+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{3\times 2\left(m+6\right)+4\times 3m}{6m\left(m+6\right)}$$

Do the multiplications in 3\times 2\left(m+6\right)+4\times 3m.

$$\frac{6m+36+12m}{6m\left(m+6\right)}$$

Combine like terms in 6m+36+12m.

$$\frac{18m+36}{6m\left(m+6\right)}$$

Factor the expressions that are not already factored in \frac{18m+36}{6m\left(m+6\right)}.

$$\frac{18\left(m+2\right)}{6m\left(m+6\right)}$$

Cancel out 6 in both numerator and denominator.

$$\frac{3\left(m+2\right)}{m\left(m+6\right)}$$

Expand m\left(m+6\right).

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

Use the distributive property to multiply 3 by m+2.

$$\frac{3m+6}{m^{2}+6m}$$

Differentiate w.r.t. m

$$-\frac{3\left(m^{2}+4m+12\right)}{\left(m\left(m+6\right)\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Factor 2m+12.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{3}{3m}+\frac{4}{2\left(m+6\right)})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3m and 2\left(m+6\right) is 6m\left(m+6\right). Multiply \frac{3}{3m} times \frac{2\left(m+6\right)}{2\left(m+6\right)}. Multiply \frac{4}{2\left(m+6\right)} times \frac{3m}{3m}.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{3\times 2\left(m+6\right)}{6m\left(m+6\right)}+\frac{4\times 3m}{6m\left(m+6\right)})$$

Since \frac{3\times 2\left(m+6\right)}{6m\left(m+6\right)} and \frac{4\times 3m}{6m\left(m+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{3\times 2\left(m+6\right)+4\times 3m}{6m\left(m+6\right)})$$

Do the multiplications in 3\times 2\left(m+6\right)+4\times 3m.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{6m+36+12m}{6m\left(m+6\right)})$$

Combine like terms in 6m+36+12m.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{18m+36}{6m\left(m+6\right)})$$

Factor the expressions that are not already factored in \frac{18m+36}{6m\left(m+6\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{18\left(m+2\right)}{6m\left(m+6\right)})$$

Cancel out 6 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{3\left(m+2\right)}{m\left(m+6\right)})$$

Use the distributive property to multiply 3 by m+2.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{3m+6}{m\left(m+6\right)})$$

Use the distributive property to multiply m by m+6.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{3m+6}{m^{2}+6m})$$

For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

$$\frac{\left(m^{2}+6m^{1}\right)\frac{\mathrm{d}}{\mathrm{d}m}(3m^{1}+6)-\left(3m^{1}+6\right)\frac{\mathrm{d}}{\mathrm{d}m}(m^{2}+6m^{1})}{\left(m^{2}+6m^{1}\right)^{2}}$$

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}.

$$\frac{\left(m^{2}+6m^{1}\right)\times 3m^{1-1}-\left(3m^{1}+6\right)\left(2m^{2-1}+6m^{1-1}\right)}{\left(m^{2}+6m^{1}\right)^{2}}$$

Simplify.

$$\frac{\left(m^{2}+6m^{1}\right)\times 3m^{0}-\left(3m^{1}+6\right)\left(2m^{1}+6m^{0}\right)}{\left(m^{2}+6m^{1}\right)^{2}}$$

Multiply m^{2}+6m^{1} times 3m^{0}.

$$\frac{m^{2}\times 3m^{0}+6m^{1}\times 3m^{0}-\left(3m^{1}+6\right)\left(2m^{1}+6m^{0}\right)}{\left(m^{2}+6m^{1}\right)^{2}}$$

Multiply 3m^{1}+6 times 2m^{1}+6m^{0}.

$$\frac{m^{2}\times 3m^{0}+6m^{1}\times 3m^{0}-\left(3m^{1}\times 2m^{1}+3m^{1}\times 6m^{0}+6\times 2m^{1}+6\times 6m^{0}\right)}{\left(m^{2}+6m^{1}\right)^{2}}$$

To multiply powers of the same base, add their exponents.

$$\frac{3m^{2}+6\times 3m^{1}-\left(3\times 2m^{1+1}+3\times 6m^{1}+6\times 2m^{1}+6\times 6m^{0}\right)}{\left(m^{2}+6m^{1}\right)^{2}}$$

Simplify.

$$\frac{3m^{2}+18m^{1}-\left(6m^{2}+18m^{1}+12m^{1}+36m^{0}\right)}{\left(m^{2}+6m^{1}\right)^{2}}$$

Combine like terms.

$$\frac{-3m^{2}-12m^{1}-36m^{0}}{\left(m^{2}+6m^{1}\right)^{2}}$$

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

$$\frac{-3m^{2}-12m-36m^{0}}{\left(m^{2}+6m\right)^{2}}$$

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

$$\frac{-3m^{2}-12m-36}{\left(m^{2}+6m\right)^{2}}$$