Solve 4w/4w^5

$$\frac { 4 w } { 4 w ^ { 5 } }$$

Evaluate

$$\frac{1}{w^{4}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(4w^{1}\right)^{1}\times \frac{1}{4w^{5}}$$

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

$$4^{1}\left(w^{1}\right)^{1}\times \frac{1}{4}\times \frac{1}{w^{5}}$$

Use the Commutative Property of Multiplication.

$$4^{1}\times \frac{1}{4}\left(w^{1}\right)^{1}\times \frac{1}{w^{5}}$$

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

$$4^{1}\times \frac{1}{4}w^{1}w^{5\left(-1\right)}$$

Multiply 5 times -1.

$$4^{1}\times \frac{1}{4}w^{1}w^{-5}$$

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

$$4^{1}\times \frac{1}{4}w^{1-5}$$

Add the exponents 1 and -5.

$$4^{1}\times \frac{1}{4}w^{-4}$$

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

$$4^{1-1}w^{-4}$$

Add the exponents 1 and -1.

$$4^{0}w^{-4}$$

For any term t except 0, t^{0}=1.

$$1w^{-4}$$

For any term t, t\times 1=t and 1t=t.

$$w^{-4}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{4^{1}w^{1}}{4^{1}w^{5}}$$

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

$$4^{1-1}w^{1-5}$$

Subtract 1 from 1.

$$4^{0}w^{1-5}$$

For any number a except 0, a^{0}=1.

$$w^{1-5}$$

Subtract 5 from 1.

$$w^{-4}$$

Differentiate w.r.t. w

$$-\frac{4}{w^{5}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{4}{4}w^{1-5})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}w}(w^{-4})$$

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}.

$$-4w^{-4-1}$$

Do the arithmetic.

$$-4w^{-5}$$