Solve 18n*100/106+18n

$$\frac { 18 n \times 100 } { 106 + 18 n }$$

Evaluate

$$\frac{900n}{9n+53}$$

Short Solution Steps

Multiply 18 and 100 to get 1800.

$$\frac{1800n}{106+18n}$$

Factor the expressions that are not already factored.

$$\frac{1800n}{2\left(9n+53\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{900n}{9n+53}$$

Differentiate w.r.t. n

$$\frac{47700}{\left(9n+53\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Multiply 18 and 100 to get 1800.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{1800n}{106+18n})$$

Factor the expressions that are not already factored in \frac{1800n}{106+18n}.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{1800n}{2\left(9n+53\right)})$$

Cancel out 2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{900n}{9n+53})$$

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.

$$\frac{\left(9n^{1}+53\right)\frac{\mathrm{d}}{\mathrm{d}n}(900n^{1})-900n^{1}\frac{\mathrm{d}}{\mathrm{d}n}(9n^{1}+53)}{\left(9n^{1}+53\right)^{2}}$$

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}.

$$\frac{\left(9n^{1}+53\right)\times 900n^{1-1}-900n^{1}\times 9n^{1-1}}{\left(9n^{1}+53\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(9n^{1}+53\right)\times 900n^{0}-900n^{1}\times 9n^{0}}{\left(9n^{1}+53\right)^{2}}$$

Expand using distributive property.

$$\frac{9n^{1}\times 900n^{0}+53\times 900n^{0}-900n^{1}\times 9n^{0}}{\left(9n^{1}+53\right)^{2}}$$

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

$$\frac{9\times 900n^{1}+53\times 900n^{0}-900\times 9n^{1}}{\left(9n^{1}+53\right)^{2}}$$

Do the arithmetic.

$$\frac{8100n^{1}+47700n^{0}-8100n^{1}}{\left(9n^{1}+53\right)^{2}}$$

Combine like terms.

$$\frac{\left(8100-8100\right)n^{1}+47700n^{0}}{\left(9n^{1}+53\right)^{2}}$$

Subtract 8100 from 8100.

$$\frac{47700n^{0}}{\left(9n^{1}+53\right)^{2}}$$

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

$$\frac{47700n^{0}}{\left(9n+53\right)^{2}}$$

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

$$\frac{47700\times 1}{\left(9n+53\right)^{2}}$$

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

$$\frac{47700}{\left(9n+53\right)^{2}}$$