Solve 7w/3w^6

$$\frac { 7 w } { 3 w ^ { 6 } }$$

Evaluate

$$\frac{7}{3w^{5}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(7w^{1}\right)^{1}\times \frac{1}{3w^{6}}$$

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

$$7^{1}\left(w^{1}\right)^{1}\times \frac{1}{3}\times \frac{1}{w^{6}}$$

Use the Commutative Property of Multiplication.

$$7^{1}\times \frac{1}{3}\left(w^{1}\right)^{1}\times \frac{1}{w^{6}}$$

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

$$7^{1}\times \frac{1}{3}w^{1}w^{6\left(-1\right)}$$

Multiply 6 times -1.

$$7^{1}\times \frac{1}{3}w^{1}w^{-6}$$

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

$$7^{1}\times \frac{1}{3}w^{1-6}$$

Add the exponents 1 and -6.

$$7^{1}\times \frac{1}{3}w^{-5}$$

Raise 7 to the power 1.

$$7\times \frac{1}{3}w^{-5}$$

Multiply 7 times \frac{1}{3}.

$$\frac{7}{3}w^{-5}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{7^{1}w^{1}}{3^{1}w^{6}}$$

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

$$\frac{7^{1}w^{1-6}}{3^{1}}$$

Subtract 6 from 1.

$$\frac{7^{1}w^{-5}}{3^{1}}$$

Divide 7 by 3.

$$\frac{7}{3}w^{-5}$$

Differentiate w.r.t. w

$$-\frac{35}{3w^{6}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{7}{3}w^{1-6})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{7}{3}w^{-5})$$

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}.

$$-5\times \frac{7}{3}w^{-5-1}$$

Do the arithmetic.

$$-\frac{35}{3}w^{-6}$$