Solve 14y/42y^2

$$\frac { 14 y } { 42 y ^ { 2 } }$$

Evaluate

$$\frac{1}{3y}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(14y^{1}\right)^{1}\times \frac{1}{42y^{2}}$$

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

$$14^{1}\left(y^{1}\right)^{1}\times \frac{1}{42}\times \frac{1}{y^{2}}$$

Use the Commutative Property of Multiplication.

$$14^{1}\times \frac{1}{42}\left(y^{1}\right)^{1}\times \frac{1}{y^{2}}$$

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

$$14^{1}\times \frac{1}{42}y^{1}y^{2\left(-1\right)}$$

Multiply 2 times -1.

$$14^{1}\times \frac{1}{42}y^{1}y^{-2}$$

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

$$14^{1}\times \frac{1}{42}y^{1-2}$$

Add the exponents 1 and -2.

$$14^{1}\times \frac{1}{42}\times \frac{1}{y}$$

Raise 14 to the power 1.

$$14\times \frac{1}{42}\times \frac{1}{y}$$

Multiply 14 times \frac{1}{42}.

$$\frac{1}{3}\times \frac{1}{y}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{14^{1}y^{1}}{42^{1}y^{2}}$$

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

$$\frac{14^{1}y^{1-2}}{42^{1}}$$

Subtract 2 from 1.

$$\frac{14^{1}\times \frac{1}{y}}{42^{1}}$$

Reduce the fraction \frac{14}{42} to lowest terms by extracting and canceling out 14.

$$\frac{1}{3}\times \frac{1}{y}$$

Differentiate w.r.t. y

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{14}{42}y^{1-2})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

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