Solve 3/5a+6+4/a-4

$$\frac { 3 } { 5 a + 6 } + \frac { 4 } { a - 4 }$$

Evaluate

$$\frac{23a+12}{\left(a-4\right)\left(5a+6\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5a+6 and a-4 is \left(a-4\right)\left(5a+6\right). Multiply \frac{3}{5a+6} times \frac{a-4}{a-4}. Multiply \frac{4}{a-4} times \frac{5a+6}{5a+6}.

$$\frac{3\left(a-4\right)}{\left(a-4\right)\left(5a+6\right)}+\frac{4\left(5a+6\right)}{\left(a-4\right)\left(5a+6\right)}$$

Since \frac{3\left(a-4\right)}{\left(a-4\right)\left(5a+6\right)} and \frac{4\left(5a+6\right)}{\left(a-4\right)\left(5a+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{3\left(a-4\right)+4\left(5a+6\right)}{\left(a-4\right)\left(5a+6\right)}$$

Do the multiplications in 3\left(a-4\right)+4\left(5a+6\right).

$$\frac{3a-12+20a+24}{\left(a-4\right)\left(5a+6\right)}$$

Combine like terms in 3a-12+20a+24.

$$\frac{23a+12}{\left(a-4\right)\left(5a+6\right)}$$

Expand \left(a-4\right)\left(5a+6\right).

$$\frac{23a+12}{5a^{2}-14a-24}$$

Differentiate w.r.t. a

$$-\frac{115a^{2}+120a+384}{\left(\left(a-4\right)\left(5a+6\right)\right)^{2}}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5a+6 and a-4 is \left(a-4\right)\left(5a+6\right). Multiply \frac{3}{5a+6} times \frac{a-4}{a-4}. Multiply \frac{4}{a-4} times \frac{5a+6}{5a+6}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3\left(a-4\right)}{\left(a-4\right)\left(5a+6\right)}+\frac{4\left(5a+6\right)}{\left(a-4\right)\left(5a+6\right)})$$

Since \frac{3\left(a-4\right)}{\left(a-4\right)\left(5a+6\right)} and \frac{4\left(5a+6\right)}{\left(a-4\right)\left(5a+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3\left(a-4\right)+4\left(5a+6\right)}{\left(a-4\right)\left(5a+6\right)})$$

Do the multiplications in 3\left(a-4\right)+4\left(5a+6\right).

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3a-12+20a+24}{\left(a-4\right)\left(5a+6\right)})$$

Combine like terms in 3a-12+20a+24.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{23a+12}{\left(a-4\right)\left(5a+6\right)})$$

Apply the distributive property by multiplying each term of a-4 by each term of 5a+6.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{23a+12}{5a^{2}+6a-20a-24})$$

Combine 6a and -20a to get -14a.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{23a+12}{5a^{2}-14a-24})$$

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.

$$\frac{\left(5a^{2}-14a^{1}-24\right)\frac{\mathrm{d}}{\mathrm{d}a}(23a^{1}+12)-\left(23a^{1}+12\right)\frac{\mathrm{d}}{\mathrm{d}a}(5a^{2}-14a^{1}-24)}{\left(5a^{2}-14a^{1}-24\right)^{2}}$$

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}.

$$\frac{\left(5a^{2}-14a^{1}-24\right)\times 23a^{1-1}-\left(23a^{1}+12\right)\left(2\times 5a^{2-1}-14a^{1-1}\right)}{\left(5a^{2}-14a^{1}-24\right)^{2}}$$

Simplify.

$$\frac{\left(5a^{2}-14a^{1}-24\right)\times 23a^{0}-\left(23a^{1}+12\right)\left(10a^{1}-14a^{0}\right)}{\left(5a^{2}-14a^{1}-24\right)^{2}}$$

Multiply 5a^{2}-14a^{1}-24 times 23a^{0}.

$$\frac{5a^{2}\times 23a^{0}-14a^{1}\times 23a^{0}-24\times 23a^{0}-\left(23a^{1}+12\right)\left(10a^{1}-14a^{0}\right)}{\left(5a^{2}-14a^{1}-24\right)^{2}}$$

Multiply 23a^{1}+12 times 10a^{1}-14a^{0}.

$$\frac{5a^{2}\times 23a^{0}-14a^{1}\times 23a^{0}-24\times 23a^{0}-\left(23a^{1}\times 10a^{1}+23a^{1}\left(-14\right)a^{0}+12\times 10a^{1}+12\left(-14\right)a^{0}\right)}{\left(5a^{2}-14a^{1}-24\right)^{2}}$$

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

$$\frac{5\times 23a^{2}-14\times 23a^{1}-24\times 23a^{0}-\left(23\times 10a^{1+1}+23\left(-14\right)a^{1}+12\times 10a^{1}+12\left(-14\right)a^{0}\right)}{\left(5a^{2}-14a^{1}-24\right)^{2}}$$

Simplify.

$$\frac{115a^{2}-322a^{1}-552a^{0}-\left(230a^{2}-322a^{1}+120a^{1}-168a^{0}\right)}{\left(5a^{2}-14a^{1}-24\right)^{2}}$$

Combine like terms.

$$\frac{-115a^{2}-120a^{1}-384a^{0}}{\left(5a^{2}-14a^{1}-24\right)^{2}}$$

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

$$\frac{-115a^{2}-120a-384a^{0}}{\left(5a^{2}-14a-24\right)^{2}}$$

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

$$\frac{-115a^{2}-120a-384}{\left(5a^{2}-14a-24\right)^{2}}$$