Solve 3m-1+19/3m+1

$$3 m - 1 + \frac { 19 } { 3 m + 1 }$$

Evaluate

$$\frac{9\left(m^{2}+2\right)}{3m+1}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 3m-1 times \frac{3m+1}{3m+1}.

$$\frac{\left(3m-1\right)\left(3m+1\right)}{3m+1}+\frac{19}{3m+1}$$

Since \frac{\left(3m-1\right)\left(3m+1\right)}{3m+1} and \frac{19}{3m+1} have the same denominator, add them by adding their numerators.

$$\frac{\left(3m-1\right)\left(3m+1\right)+19}{3m+1}$$

Do the multiplications in \left(3m-1\right)\left(3m+1\right)+19.

$$\frac{9m^{2}+3m-3m-1+19}{3m+1}$$

Combine like terms in 9m^{2}+3m-3m-1+19.

$$\frac{9m^{2}+18}{3m+1}$$

Differentiate w.r.t. m

$$\frac{9\left(3m^{2}+2m-6\right)}{\left(3m+1\right)^{2}}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Multiply 3m-1 times \frac{3m+1}{3m+1}.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{\left(3m-1\right)\left(3m+1\right)}{3m+1}+\frac{19}{3m+1})$$

Since \frac{\left(3m-1\right)\left(3m+1\right)}{3m+1} and \frac{19}{3m+1} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{\left(3m-1\right)\left(3m+1\right)+19}{3m+1})$$

Do the multiplications in \left(3m-1\right)\left(3m+1\right)+19.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{9m^{2}+3m-3m-1+19}{3m+1})$$

Combine like terms in 9m^{2}+3m-3m-1+19.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{9m^{2}+18}{3m+1})$$

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.

$$\frac{\left(3m^{1}+1\right)\frac{\mathrm{d}}{\mathrm{d}m}(9m^{2}+18)-\left(9m^{2}+18\right)\frac{\mathrm{d}}{\mathrm{d}m}(3m^{1}+1)}{\left(3m^{1}+1\right)^{2}}$$

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}.

$$\frac{\left(3m^{1}+1\right)\times 2\times 9m^{2-1}-\left(9m^{2}+18\right)\times 3m^{1-1}}{\left(3m^{1}+1\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(3m^{1}+1\right)\times 18m^{1}-\left(9m^{2}+18\right)\times 3m^{0}}{\left(3m^{1}+1\right)^{2}}$$

Expand using distributive property.

$$\frac{3m^{1}\times 18m^{1}+18m^{1}-\left(9m^{2}\times 3m^{0}+18\times 3m^{0}\right)}{\left(3m^{1}+1\right)^{2}}$$

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

$$\frac{3\times 18m^{1+1}+18m^{1}-\left(9\times 3m^{2}+18\times 3m^{0}\right)}{\left(3m^{1}+1\right)^{2}}$$

Do the arithmetic.

$$\frac{54m^{2}+18m^{1}-\left(27m^{2}+54m^{0}\right)}{\left(3m^{1}+1\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{54m^{2}+18m^{1}-27m^{2}-54m^{0}}{\left(3m^{1}+1\right)^{2}}$$

Combine like terms.

$$\frac{\left(54-27\right)m^{2}+18m^{1}-54m^{0}}{\left(3m^{1}+1\right)^{2}}$$

Subtract 27 from 54.

$$\frac{27m^{2}+18m^{1}-54m^{0}}{\left(3m^{1}+1\right)^{2}}$$

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

$$\frac{27m^{2}+18m-54m^{0}}{\left(3m+1\right)^{2}}$$

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

$$\frac{27m^{2}+18m-54\times 1}{\left(3m+1\right)^{2}}$$

For any term t, t\times 1=t and 1t=t.

$$\frac{27m^{2}+18m-54}{\left(3m+1\right)^{2}}$$