Solve -19/1-28z-45000

$$\frac { - 19 } { 1 - 28 z } - 45000$$

Evaluate

$$\frac{1260000z-45019}{1-28z}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 45000 times \frac{1-28z}{1-28z}.

$$\frac{-19}{1-28z}-\frac{45000\left(1-28z\right)}{1-28z}$$

Since \frac{-19}{1-28z} and \frac{45000\left(1-28z\right)}{1-28z} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-19-45000\left(1-28z\right)}{1-28z}$$

Do the multiplications in -19-45000\left(1-28z\right).

$$\frac{-19-45000+1260000z}{1-28z}$$

Combine like terms in -19-45000+1260000z.

$$\frac{-45019+1260000z}{1-28z}$$

Differentiate w.r.t. z

$$-\frac{532}{\left(28z-1\right)^{2}}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Multiply 45000 times \frac{1-28z}{1-28z}.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{-19}{1-28z}-\frac{45000\left(1-28z\right)}{1-28z})$$

Since \frac{-19}{1-28z} and \frac{45000\left(1-28z\right)}{1-28z} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{-19-45000\left(1-28z\right)}{1-28z})$$

Do the multiplications in -19-45000\left(1-28z\right).

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{-19-45000+1260000z}{1-28z})$$

Combine like terms in -19-45000+1260000z.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{-45019+1260000z}{1-28z})$$

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.

$$\frac{\left(-28z^{1}+1\right)\frac{\mathrm{d}}{\mathrm{d}z}(1260000z^{1}-45019)-\left(1260000z^{1}-45019\right)\frac{\mathrm{d}}{\mathrm{d}z}(-28z^{1}+1)}{\left(-28z^{1}+1\right)^{2}}$$

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}.

$$\frac{\left(-28z^{1}+1\right)\times 1260000z^{1-1}-\left(1260000z^{1}-45019\right)\left(-28\right)z^{1-1}}{\left(-28z^{1}+1\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(-28z^{1}+1\right)\times 1260000z^{0}-\left(1260000z^{1}-45019\right)\left(-28\right)z^{0}}{\left(-28z^{1}+1\right)^{2}}$$

Expand using distributive property.

$$\frac{-28z^{1}\times 1260000z^{0}+1260000z^{0}-\left(1260000z^{1}\left(-28\right)z^{0}-45019\left(-28\right)z^{0}\right)}{\left(-28z^{1}+1\right)^{2}}$$

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

$$\frac{-28\times 1260000z^{1}+1260000z^{0}-\left(1260000\left(-28\right)z^{1}-45019\left(-28\right)z^{0}\right)}{\left(-28z^{1}+1\right)^{2}}$$

Do the arithmetic.

$$\frac{-35280000z^{1}+1260000z^{0}-\left(-35280000z^{1}+1260532z^{0}\right)}{\left(-28z^{1}+1\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{-35280000z^{1}+1260000z^{0}-\left(-35280000z^{1}\right)-1260532z^{0}}{\left(-28z^{1}+1\right)^{2}}$$

Combine like terms.

$$\frac{\left(-35280000-\left(-35280000\right)\right)z^{1}+\left(1260000-1260532\right)z^{0}}{\left(-28z^{1}+1\right)^{2}}$$

Subtract -35280000 from -35280000 and 1260532 from 1260000.

$$\frac{-532z^{0}}{\left(-28z^{1}+1\right)^{2}}$$

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

$$\frac{-532z^{0}}{\left(-28z+1\right)^{2}}$$

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

$$\frac{-532}{\left(-28z+1\right)^{2}}$$