Solve 15n/30n^3

$$\frac { 15 n } { 30 n ^ { 3 } }$$

Evaluate

$$\frac{1}{2n^{2}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(15n^{1}\right)^{1}\times \frac{1}{30n^{3}}$$

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

$$15^{1}\left(n^{1}\right)^{1}\times \frac{1}{30}\times \frac{1}{n^{3}}$$

Use the Commutative Property of Multiplication.

$$15^{1}\times \frac{1}{30}\left(n^{1}\right)^{1}\times \frac{1}{n^{3}}$$

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

$$15^{1}\times \frac{1}{30}n^{1}n^{3\left(-1\right)}$$

Multiply 3 times -1.

$$15^{1}\times \frac{1}{30}n^{1}n^{-3}$$

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

$$15^{1}\times \frac{1}{30}n^{1-3}$$

Add the exponents 1 and -3.

$$15^{1}\times \frac{1}{30}n^{-2}$$

Raise 15 to the power 1.

$$15\times \frac{1}{30}n^{-2}$$

Multiply 15 times \frac{1}{30}.

$$\frac{1}{2}n^{-2}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{15^{1}n^{1}}{30^{1}n^{3}}$$

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

$$\frac{15^{1}n^{1-3}}{30^{1}}$$

Subtract 3 from 1.

$$\frac{15^{1}n^{-2}}{30^{1}}$$

Reduce the fraction \frac{15}{30} to lowest terms by extracting and canceling out 15.

$$\frac{1}{2}n^{-2}$$

Differentiate w.r.t. n

$$-\frac{1}{n^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{15}{30}n^{1-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{1}{2}n^{-2})$$

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}.

$$-2\times \frac{1}{2}n^{-2-1}$$

Do the arithmetic.

$$-n^{-3}$$