Solve 3n/2n^2

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

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Use the Commutative Property of Multiplication.

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

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

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

Multiply 2 times -1.

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

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

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

Add the exponents 1 and -2.

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

Raise 3 to the power 1.

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

Multiply 3 times \frac{1}{2}.

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

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Subtract 2 from 1.

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

Divide 3 by 2.

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

Differentiate w.r.t. n

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

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{3}{2}\times \frac{1}{n})$$

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}.

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

Do the arithmetic.

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