Solve 2N/3N^2

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

Evaluate

$$\frac{2}{3N}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Use the Commutative Property of Multiplication.

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

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

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

Multiply 2 times -1.

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

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

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

Add the exponents 1 and -2.

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

Raise 2 to the power 1.

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

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

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

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Subtract 2 from 1.

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

Divide 2 by 3.

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

Differentiate w.r.t. N

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

Solution Steps

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

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

Do the arithmetic.

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

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}.

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

Do the arithmetic.

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