Solve 5n/10n-6=

$$\frac { 5 n } { 10 n - 6 } =$$

Differentiate w.r.t. n

$$-\frac{15}{2\left(5n-3\right)^{2}}$$

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.

$$\frac{\left(10n^{1}-6\right)\frac{\mathrm{d}}{\mathrm{d}n}(5n^{1})-5n^{1}\frac{\mathrm{d}}{\mathrm{d}n}(10n^{1}-6)}{\left(10n^{1}-6\right)^{2}}$$

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}.

$$\frac{\left(10n^{1}-6\right)\times 5n^{1-1}-5n^{1}\times 10n^{1-1}}{\left(10n^{1}-6\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(10n^{1}-6\right)\times 5n^{0}-5n^{1}\times 10n^{0}}{\left(10n^{1}-6\right)^{2}}$$

Expand using distributive property.

$$\frac{10n^{1}\times 5n^{0}-6\times 5n^{0}-5n^{1}\times 10n^{0}}{\left(10n^{1}-6\right)^{2}}$$

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

$$\frac{10\times 5n^{1}-6\times 5n^{0}-5\times 10n^{1}}{\left(10n^{1}-6\right)^{2}}$$

Do the arithmetic.

$$\frac{50n^{1}-30n^{0}-50n^{1}}{\left(10n^{1}-6\right)^{2}}$$

Combine like terms.

$$\frac{\left(50-50\right)n^{1}-30n^{0}}{\left(10n^{1}-6\right)^{2}}$$

Subtract 50 from 50.

$$\frac{-30n^{0}}{\left(10n^{1}-6\right)^{2}}$$

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

$$\frac{-30n^{0}}{\left(10n-6\right)^{2}}$$

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

$$\frac{-30}{\left(10n-6\right)^{2}}$$

Evaluate

$$\frac{5n}{2\left(5n-3\right)}$$