Solve 3n/3n^3

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

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Use the Commutative Property of Multiplication.

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

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

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

Multiply 3 times -1.

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

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

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

Add the exponents 1 and -3.

$$3^{1}\times \frac{1}{3}n^{-2}$$

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

$$3^{1-1}n^{-2}$$

Add the exponents 1 and -1.

$$3^{0}n^{-2}$$

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

$$1n^{-2}$$

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

$$n^{-2}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Subtract 1 from 1.

$$3^{0}n^{1-3}$$

For any number a except 0, a^{0}=1.

$$n^{1-3}$$

Subtract 3 from 1.

$$n^{-2}$$

Differentiate w.r.t. n

$$-\frac{2}{n^{3}}$$

Solution Steps

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

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

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$-2n^{-3}$$