Solve (-1/5y^3)(1/3y^9)

$$( - \frac { 1 } { 5 } y ^ { 3 } ) ( \frac { 1 } { 3 } y ^ { 9 } )$$

Evaluate

$$-\frac{y^{12}}{15}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 9 to get 12.

$$-\frac{1}{5}y^{12}\times \frac{1}{3}$$

Multiply -\frac{1}{5} and \frac{1}{3} to get -\frac{1}{15}.

$$-\frac{1}{15}y^{12}$$

Differentiate w.r.t. y

$$-\frac{4y^{11}}{5}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 3 and 9 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}y}(-\frac{1}{5}y^{12}\times \frac{1}{3})$$

Multiply -\frac{1}{5} and \frac{1}{3} to get -\frac{1}{15}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(-\frac{1}{15}y^{12})$$

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

$$12\left(-\frac{1}{15}\right)y^{12-1}$$

Multiply 12 times -\frac{1}{15}.

$$-\frac{4}{5}y^{12-1}$$

Subtract 1 from 12.

$$-\frac{4}{5}y^{11}$$