Solve 3z/12z^2

$$\frac { 3 z } { 12 z ^ { 2 } }$$

Evaluate

$$\frac{1}{4z}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(3z^{1}\right)^{1}\times \frac{1}{12z^{2}}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$3^{1}\left(z^{1}\right)^{1}\times \frac{1}{12}\times \frac{1}{z^{2}}$$

Use the Commutative Property of Multiplication.

$$3^{1}\times \frac{1}{12}\left(z^{1}\right)^{1}\times \frac{1}{z^{2}}$$

To raise a power to another power, multiply the exponents.

$$3^{1}\times \frac{1}{12}z^{1}z^{2\left(-1\right)}$$

Multiply 2 times -1.

$$3^{1}\times \frac{1}{12}z^{1}z^{-2}$$

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

$$3^{1}\times \frac{1}{12}z^{1-2}$$

Add the exponents 1 and -2.

$$3^{1}\times \frac{1}{12}\times \frac{1}{z}$$

Raise 3 to the power 1.

$$3\times \frac{1}{12}\times \frac{1}{z}$$

Multiply 3 times \frac{1}{12}.

$$\frac{1}{4}\times \frac{1}{z}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{3^{1}z^{1}}{12^{1}z^{2}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{3^{1}z^{1-2}}{12^{1}}$$

Subtract 2 from 1.

$$\frac{3^{1}\times \frac{1}{z}}{12^{1}}$$

Reduce the fraction \frac{3}{12} to lowest terms by extracting and canceling out 3.

$$\frac{1}{4}\times \frac{1}{z}$$

Differentiate w.r.t. z

$$-\frac{1}{4z^{2}}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{3}{12}z^{1-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{1}{4}\times \frac{1}{z})$$

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}.

$$-\frac{1}{4}z^{-1-1}$$

Do the arithmetic.

$$-\frac{1}{4}z^{-2}$$