Solve 2/15ab^3:(-4/5ab)

$$\frac { 2 } { 15 } a b ^ { 3 } : ( - \frac { 4 } { 5 } a b )$$

Evaluate

$$-\frac{b^{2}}{6}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{\left(\frac{2}{15}\right)^{1}a^{1}b^{3}}{\left(-\frac{4}{5}\right)^{1}a^{1}b^{1}}$$

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

$$\frac{\left(\frac{2}{15}\right)^{1}}{\left(-\frac{4}{5}\right)^{1}}a^{1-1}b^{3-1}$$

Subtract 1 from 1.

$$\frac{\left(\frac{2}{15}\right)^{1}}{\left(-\frac{4}{5}\right)^{1}}a^{0}b^{3-1}$$

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

$$\frac{\left(\frac{2}{15}\right)^{1}}{\left(-\frac{4}{5}\right)^{1}}b^{3-1}$$

Subtract 1 from 3.

$$\frac{\left(\frac{2}{15}\right)^{1}}{\left(-\frac{4}{5}\right)^{1}}b^{2}$$

Divide \frac{2}{15} by -\frac{4}{5} by multiplying \frac{2}{15} by the reciprocal of -\frac{4}{5}.

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

Differentiate w.r.t. b

$$-\frac{b}{3}$$

Steps Using Definition of a Derivative

Cancel out ab in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{\frac{2}{15}b^{2}}{-\frac{4}{5}})$$

Divide \frac{2}{15}b^{2} by -\frac{4}{5} by multiplying \frac{2}{15}b^{2} by the reciprocal of -\frac{4}{5}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{\frac{2}{15}b^{2}\times 5}{-4})$$

Multiply \frac{2}{15} and 5 to get \frac{2}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{\frac{2}{3}b^{2}}{-4})$$

Divide \frac{2}{3}b^{2} by -4 to get -\frac{1}{6}b^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(-\frac{1}{6}b^{2})$$

The derivative of ax^{n} is nax^{n-1}.

$$2\left(-\frac{1}{6}\right)b^{2-1}$$

Multiply 2 times -\frac{1}{6}.

$$-\frac{1}{3}b^{2-1}$$

Subtract 1 from 2.

$$-\frac{1}{3}b^{1}$$

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

$$-\frac{1}{3}b$$