Solve 2v/3v-9

$$\frac { 2 v } { 3 v - 9 }$$

Differentiate w.r.t. v

$$-\frac{2}{\left(v-3\right)^{2}}$$

Steps Using Derivative Rule for Quotient

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.

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

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}.

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

Do the arithmetic.

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

Expand using distributive property.

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

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

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

Do the arithmetic.

$$\frac{6v^{1}-18v^{0}-6v^{1}}{\left(3v^{1}-9\right)^{2}}$$

Combine like terms.

$$\frac{\left(6-6\right)v^{1}-18v^{0}}{\left(3v^{1}-9\right)^{2}}$$

Subtract 6 from 6.

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

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

$$\frac{-18v^{0}}{\left(3v-9\right)^{2}}$$

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

$$\frac{-18}{\left(3v-9\right)^{2}}$$

Evaluate

$$\frac{2v}{3\left(v-3\right)}$$