Solve 5/3m+2+1/m+5=

$$\frac { 5 } { 3 m + 2 } + \frac { 1 } { m + 5 } =$$

Evaluate

$$\frac{8m+27}{\left(m+5\right)\left(3m+2\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3m+2 and m+5 is \left(m+5\right)\left(3m+2\right). Multiply \frac{5}{3m+2} times \frac{m+5}{m+5}. Multiply \frac{1}{m+5} times \frac{3m+2}{3m+2}.

$$\frac{5\left(m+5\right)}{\left(m+5\right)\left(3m+2\right)}+\frac{3m+2}{\left(m+5\right)\left(3m+2\right)}$$

Since \frac{5\left(m+5\right)}{\left(m+5\right)\left(3m+2\right)} and \frac{3m+2}{\left(m+5\right)\left(3m+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{5\left(m+5\right)+3m+2}{\left(m+5\right)\left(3m+2\right)}$$

Do the multiplications in 5\left(m+5\right)+3m+2.

$$\frac{5m+25+3m+2}{\left(m+5\right)\left(3m+2\right)}$$

Combine like terms in 5m+25+3m+2.

$$\frac{8m+27}{\left(m+5\right)\left(3m+2\right)}$$

Expand \left(m+5\right)\left(3m+2\right).

$$\frac{8m+27}{3m^{2}+17m+10}$$

Differentiate w.r.t. m

$$-\frac{24m^{2}+162m+379}{\left(\left(m+5\right)\left(3m+2\right)\right)^{2}}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3m+2 and m+5 is \left(m+5\right)\left(3m+2\right). Multiply \frac{5}{3m+2} times \frac{m+5}{m+5}. Multiply \frac{1}{m+5} times \frac{3m+2}{3m+2}.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{5\left(m+5\right)}{\left(m+5\right)\left(3m+2\right)}+\frac{3m+2}{\left(m+5\right)\left(3m+2\right)})$$

Since \frac{5\left(m+5\right)}{\left(m+5\right)\left(3m+2\right)} and \frac{3m+2}{\left(m+5\right)\left(3m+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{5\left(m+5\right)+3m+2}{\left(m+5\right)\left(3m+2\right)})$$

Do the multiplications in 5\left(m+5\right)+3m+2.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{5m+25+3m+2}{\left(m+5\right)\left(3m+2\right)})$$

Combine like terms in 5m+25+3m+2.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{8m+27}{\left(m+5\right)\left(3m+2\right)})$$

Apply the distributive property by multiplying each term of m+5 by each term of 3m+2.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{8m+27}{3m^{2}+2m+15m+10})$$

Combine 2m and 15m to get 17m.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{8m+27}{3m^{2}+17m+10})$$

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.

$$\frac{\left(3m^{2}+17m^{1}+10\right)\frac{\mathrm{d}}{\mathrm{d}m}(8m^{1}+27)-\left(8m^{1}+27\right)\frac{\mathrm{d}}{\mathrm{d}m}(3m^{2}+17m^{1}+10)}{\left(3m^{2}+17m^{1}+10\right)^{2}}$$

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}.

$$\frac{\left(3m^{2}+17m^{1}+10\right)\times 8m^{1-1}-\left(8m^{1}+27\right)\left(2\times 3m^{2-1}+17m^{1-1}\right)}{\left(3m^{2}+17m^{1}+10\right)^{2}}$$

Simplify.

$$\frac{\left(3m^{2}+17m^{1}+10\right)\times 8m^{0}-\left(8m^{1}+27\right)\left(6m^{1}+17m^{0}\right)}{\left(3m^{2}+17m^{1}+10\right)^{2}}$$

Multiply 3m^{2}+17m^{1}+10 times 8m^{0}.

$$\frac{3m^{2}\times 8m^{0}+17m^{1}\times 8m^{0}+10\times 8m^{0}-\left(8m^{1}+27\right)\left(6m^{1}+17m^{0}\right)}{\left(3m^{2}+17m^{1}+10\right)^{2}}$$

Multiply 8m^{1}+27 times 6m^{1}+17m^{0}.

$$\frac{3m^{2}\times 8m^{0}+17m^{1}\times 8m^{0}+10\times 8m^{0}-\left(8m^{1}\times 6m^{1}+8m^{1}\times 17m^{0}+27\times 6m^{1}+27\times 17m^{0}\right)}{\left(3m^{2}+17m^{1}+10\right)^{2}}$$

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

$$\frac{3\times 8m^{2}+17\times 8m^{1}+10\times 8m^{0}-\left(8\times 6m^{1+1}+8\times 17m^{1}+27\times 6m^{1}+27\times 17m^{0}\right)}{\left(3m^{2}+17m^{1}+10\right)^{2}}$$

Simplify.

$$\frac{24m^{2}+136m^{1}+80m^{0}-\left(48m^{2}+136m^{1}+162m^{1}+459m^{0}\right)}{\left(3m^{2}+17m^{1}+10\right)^{2}}$$

Combine like terms.

$$\frac{-24m^{2}-162m^{1}-379m^{0}}{\left(3m^{2}+17m^{1}+10\right)^{2}}$$

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

$$\frac{-24m^{2}-162m-379m^{0}}{\left(3m^{2}+17m+10\right)^{2}}$$

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

$$\frac{-24m^{2}-162m-379}{\left(3m^{2}+17m+10\right)^{2}}$$