Solve 5/3k+2-4/2k+7

$$\frac { 5 } { 3 k + 2 } - \frac { 4 } { 2 k + 7 }$$

Evaluate

$$\frac{27-2k}{\left(2k+7\right)\left(3k+2\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3k+2 and 2k+7 is \left(2k+7\right)\left(3k+2\right). Multiply \frac{5}{3k+2} times \frac{2k+7}{2k+7}. Multiply \frac{4}{2k+7} times \frac{3k+2}{3k+2}.

$$\frac{5\left(2k+7\right)}{\left(2k+7\right)\left(3k+2\right)}-\frac{4\left(3k+2\right)}{\left(2k+7\right)\left(3k+2\right)}$$

Since \frac{5\left(2k+7\right)}{\left(2k+7\right)\left(3k+2\right)} and \frac{4\left(3k+2\right)}{\left(2k+7\right)\left(3k+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5\left(2k+7\right)-4\left(3k+2\right)}{\left(2k+7\right)\left(3k+2\right)}$$

Do the multiplications in 5\left(2k+7\right)-4\left(3k+2\right).

$$\frac{10k+35-12k-8}{\left(2k+7\right)\left(3k+2\right)}$$

Combine like terms in 10k+35-12k-8.

$$\frac{-2k+27}{\left(2k+7\right)\left(3k+2\right)}$$

Expand \left(2k+7\right)\left(3k+2\right).

$$\frac{-2k+27}{6k^{2}+25k+14}$$

Differentiate w.r.t. k

$$\frac{12k^{2}-324k-703}{36k^{4}+300k^{3}+793k^{2}+700k+196}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3k+2 and 2k+7 is \left(2k+7\right)\left(3k+2\right). Multiply \frac{5}{3k+2} times \frac{2k+7}{2k+7}. Multiply \frac{4}{2k+7} times \frac{3k+2}{3k+2}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{5\left(2k+7\right)}{\left(2k+7\right)\left(3k+2\right)}-\frac{4\left(3k+2\right)}{\left(2k+7\right)\left(3k+2\right)})$$

Since \frac{5\left(2k+7\right)}{\left(2k+7\right)\left(3k+2\right)} and \frac{4\left(3k+2\right)}{\left(2k+7\right)\left(3k+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{5\left(2k+7\right)-4\left(3k+2\right)}{\left(2k+7\right)\left(3k+2\right)})$$

Do the multiplications in 5\left(2k+7\right)-4\left(3k+2\right).

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{10k+35-12k-8}{\left(2k+7\right)\left(3k+2\right)})$$

Combine like terms in 10k+35-12k-8.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{-2k+27}{\left(2k+7\right)\left(3k+2\right)})$$

Apply the distributive property by multiplying each term of 2k+7 by each term of 3k+2.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{-2k+27}{6k^{2}+4k+21k+14})$$

Combine 4k and 21k to get 25k.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{-2k+27}{6k^{2}+25k+14})$$

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.

$$\frac{\left(6k^{2}+25k^{1}+14\right)\frac{\mathrm{d}}{\mathrm{d}k}(-2k^{1}+27)-\left(-2k^{1}+27\right)\frac{\mathrm{d}}{\mathrm{d}k}(6k^{2}+25k^{1}+14)}{\left(6k^{2}+25k^{1}+14\right)^{2}}$$

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}.

$$\frac{\left(6k^{2}+25k^{1}+14\right)\left(-2\right)k^{1-1}-\left(-2k^{1}+27\right)\left(2\times 6k^{2-1}+25k^{1-1}\right)}{\left(6k^{2}+25k^{1}+14\right)^{2}}$$

Simplify.

$$\frac{\left(6k^{2}+25k^{1}+14\right)\left(-2\right)k^{0}-\left(-2k^{1}+27\right)\left(12k^{1}+25k^{0}\right)}{\left(6k^{2}+25k^{1}+14\right)^{2}}$$

Multiply 6k^{2}+25k^{1}+14 times -2k^{0}.

$$\frac{6k^{2}\left(-2\right)k^{0}+25k^{1}\left(-2\right)k^{0}+14\left(-2\right)k^{0}-\left(-2k^{1}+27\right)\left(12k^{1}+25k^{0}\right)}{\left(6k^{2}+25k^{1}+14\right)^{2}}$$

Multiply -2k^{1}+27 times 12k^{1}+25k^{0}.

$$\frac{6k^{2}\left(-2\right)k^{0}+25k^{1}\left(-2\right)k^{0}+14\left(-2\right)k^{0}-\left(-2k^{1}\times 12k^{1}-2k^{1}\times 25k^{0}+27\times 12k^{1}+27\times 25k^{0}\right)}{\left(6k^{2}+25k^{1}+14\right)^{2}}$$

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

$$\frac{6\left(-2\right)k^{2}+25\left(-2\right)k^{1}+14\left(-2\right)k^{0}-\left(-2\times 12k^{1+1}-2\times 25k^{1}+27\times 12k^{1}+27\times 25k^{0}\right)}{\left(6k^{2}+25k^{1}+14\right)^{2}}$$

Simplify.

$$\frac{-12k^{2}-50k^{1}-28k^{0}-\left(-24k^{2}-50k^{1}+324k^{1}+675k^{0}\right)}{\left(6k^{2}+25k^{1}+14\right)^{2}}$$

Combine like terms.

$$\frac{12k^{2}-324k^{1}-703k^{0}}{\left(6k^{2}+25k^{1}+14\right)^{2}}$$

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

$$\frac{12k^{2}-324k-703k^{0}}{\left(6k^{2}+25k+14\right)^{2}}$$

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

$$\frac{12k^{2}-324k-703}{\left(6k^{2}+25k+14\right)^{2}}$$