Solve 6(y^4)^5/12y^4y^5

$$\frac { 6 ( y ^ { 4 } ) ^ { 5 } } { 12 y ^ { 4 } y ^ { 5 } }$$

Evaluate

$$\frac{y^{11}}{2}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 4 and 5 to get 20.

$$\frac{6y^{20}}{12y^{4}y^{5}}$$

To multiply powers of the same base, add their exponents. Add 4 and 5 to get 9.

$$\frac{6y^{20}}{12y^{9}}$$

Cancel out 6y^{9} in both numerator and denominator.

$$\frac{y^{11}}{2}$$

Differentiate w.r.t. y

$$\frac{11y^{10}}{2}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 4 and 5 to get 20.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{6y^{20}}{12y^{4}y^{5}})$$

To multiply powers of the same base, add their exponents. Add 4 and 5 to get 9.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{6y^{20}}{12y^{9}})$$

Cancel out 6y^{9} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{11}}{2})$$

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

$$11\times \frac{1}{2}y^{11-1}$$

Multiply 11 times \frac{1}{2}.

$$\frac{11}{2}y^{11-1}$$

Subtract 1 from 11.

$$\frac{11}{2}y^{10}$$