Solve 4y^4/36y^6

$$\frac { 4 y ^ { 4 } } { 36 y ^ { 6 } }$$

Evaluate

$$\frac{1}{9y^{2}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(4y^{4}\right)^{1}\times \frac{1}{36y^{6}}$$

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

$$4^{1}\left(y^{4}\right)^{1}\times \frac{1}{36}\times \frac{1}{y^{6}}$$

Use the Commutative Property of Multiplication.

$$4^{1}\times \frac{1}{36}\left(y^{4}\right)^{1}\times \frac{1}{y^{6}}$$

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

$$4^{1}\times \frac{1}{36}y^{4}y^{6\left(-1\right)}$$

Multiply 6 times -1.

$$4^{1}\times \frac{1}{36}y^{4}y^{-6}$$

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

$$4^{1}\times \frac{1}{36}y^{4-6}$$

Add the exponents 4 and -6.

$$4^{1}\times \frac{1}{36}y^{-2}$$

Raise 4 to the power 1.

$$4\times \frac{1}{36}y^{-2}$$

Multiply 4 times \frac{1}{36}.

$$\frac{1}{9}y^{-2}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{4^{1}y^{4}}{36^{1}y^{6}}$$

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

$$\frac{4^{1}y^{4-6}}{36^{1}}$$

Subtract 6 from 4.

$$\frac{4^{1}y^{-2}}{36^{1}}$$

Reduce the fraction \frac{4}{36} to lowest terms by extracting and canceling out 4.

$$\frac{1}{9}y^{-2}$$

Differentiate w.r.t. y

$$-\frac{2}{9y^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{4}{36}y^{4-6})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{9}y^{-2})$$

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}.

$$-2\times \frac{1}{9}y^{-2-1}$$

Do the arithmetic.

$$-\frac{2}{9}y^{-3}$$