Solve 7v/7v^4

$$\frac { 7 v } { 7 v ^ { 4 } }$$

Evaluate

$$\frac{1}{v^{3}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(7v^{1}\right)^{1}\times \frac{1}{7v^{4}}$$

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

$$7^{1}\left(v^{1}\right)^{1}\times \frac{1}{7}\times \frac{1}{v^{4}}$$

Use the Commutative Property of Multiplication.

$$7^{1}\times \frac{1}{7}\left(v^{1}\right)^{1}\times \frac{1}{v^{4}}$$

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

$$7^{1}\times \frac{1}{7}v^{1}v^{4\left(-1\right)}$$

Multiply 4 times -1.

$$7^{1}\times \frac{1}{7}v^{1}v^{-4}$$

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

$$7^{1}\times \frac{1}{7}v^{1-4}$$

Add the exponents 1 and -4.

$$7^{1}\times \frac{1}{7}v^{-3}$$

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

$$7^{1-1}v^{-3}$$

Add the exponents 1 and -1.

$$7^{0}v^{-3}$$

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

$$1v^{-3}$$

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

$$v^{-3}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{7^{1}v^{1}}{7^{1}v^{4}}$$

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

$$7^{1-1}v^{1-4}$$

Subtract 1 from 1.

$$7^{0}v^{1-4}$$

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

$$v^{1-4}$$

Subtract 4 from 1.

$$v^{-3}$$

Differentiate w.r.t. v

$$-\frac{3}{v^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{7}{7}v^{1-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}v}(v^{-3})$$

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}.

$$-3v^{-3-1}$$

Do the arithmetic.

$$-3v^{-4}$$