Solve 7v^2/42v^3

$$\frac { 7 v ^ { 2 } } { 42 v ^ { 3 } }$$

Evaluate

$$\frac{1}{6v}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(7v^{2}\right)^{1}\times \frac{1}{42v^{3}}$$

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

$$7^{1}\left(v^{2}\right)^{1}\times \frac{1}{42}\times \frac{1}{v^{3}}$$

Use the Commutative Property of Multiplication.

$$7^{1}\times \frac{1}{42}\left(v^{2}\right)^{1}\times \frac{1}{v^{3}}$$

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

$$7^{1}\times \frac{1}{42}v^{2}v^{3\left(-1\right)}$$

Multiply 3 times -1.

$$7^{1}\times \frac{1}{42}v^{2}v^{-3}$$

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

$$7^{1}\times \frac{1}{42}v^{2-3}$$

Add the exponents 2 and -3.

$$7^{1}\times \frac{1}{42}\times \frac{1}{v}$$

Raise 7 to the power 1.

$$7\times \frac{1}{42}\times \frac{1}{v}$$

Multiply 7 times \frac{1}{42}.

$$\frac{1}{6}\times \frac{1}{v}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{7^{1}v^{2}}{42^{1}v^{3}}$$

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

$$\frac{7^{1}v^{2-3}}{42^{1}}$$

Subtract 3 from 2.

$$\frac{7^{1}\times \frac{1}{v}}{42^{1}}$$

Reduce the fraction \frac{7}{42} to lowest terms by extracting and canceling out 7.

$$\frac{1}{6}\times \frac{1}{v}$$

Differentiate w.r.t. v

$$-\frac{1}{6v^{2}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{7}{42}v^{2-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{1}{6}\times \frac{1}{v})$$

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}.

$$-\frac{1}{6}v^{-1-1}$$

Do the arithmetic.

$$-\frac{1}{6}v^{-2}$$