Solve -14y^-2/42y^4

$$\frac { - 14 y ^ { - 2 } } { 42 y ^ { 4 } }$$

Evaluate

$$-\frac{1}{3y^{6}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(-14y^{-2}\right)^{1}\times \frac{1}{42y^{4}}$$

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

$$\left(-14\right)^{1}\left(y^{-2}\right)^{1}\times \frac{1}{42}\times \frac{1}{y^{4}}$$

Use the Commutative Property of Multiplication.

$$\left(-14\right)^{1}\times \frac{1}{42}\left(y^{-2}\right)^{1}\times \frac{1}{y^{4}}$$

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

$$\left(-14\right)^{1}\times \frac{1}{42}y^{-2}y^{4\left(-1\right)}$$

Multiply 4 times -1.

$$\left(-14\right)^{1}\times \frac{1}{42}y^{-2}y^{-4}$$

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

$$\left(-14\right)^{1}\times \frac{1}{42}y^{-2-4}$$

Add the exponents -2 and -4.

$$\left(-14\right)^{1}\times \frac{1}{42}y^{-6}$$

Raise -14 to the power 1.

$$-14\times \frac{1}{42}y^{-6}$$

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

$$-\frac{1}{3}y^{-6}$$

Differentiate w.r.t. y

$$\frac{2}{y^{7}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\left(-\frac{14}{42}\right)y^{-2-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}y}(-\frac{1}{3}y^{-6})$$

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}.

$$-6\left(-\frac{1}{3}\right)y^{-6-1}$$

Do the arithmetic.

$$2y^{-7}$$