Solve (frac{yz^frac{1{2}}}{64yz^-6})^frac{1{6}}

$$( \frac { y z ^ { \frac { 1 } { 2 } } } { 64 y z ^ { - 6 } } ) ^ { \frac { 1 } { 6 } }$$

Differentiate w.r.t. z

$$\frac{13\sqrt[12]{z}}{24}$$

Steps Using Definition of a Derivative

Cancel out y in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\left(\frac{\sqrt{z}}{64z^{-6}}\right)^{\frac{1}{6}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(\left(\frac{z^{\frac{13}{2}}}{64}\right)^{\frac{1}{6}})$$

To raise \frac{z^{\frac{13}{2}}}{64} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{\left(z^{\frac{13}{2}}\right)^{\frac{1}{6}}}{64^{\frac{1}{6}}})$$

To raise a power to another power, multiply the exponents. Multiply \frac{13}{2} and \frac{1}{6} to get \frac{13}{12}.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{\frac{13}{12}}}{64^{\frac{1}{6}}})$$

Calculate 64 to the power of \frac{1}{6} and get 2.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{\frac{13}{12}}}{2})$$

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

$$\frac{13}{12}\times \frac{1}{2}z^{\frac{13}{12}-1}$$

Multiply \frac{13}{12} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$\frac{13}{24}z^{\frac{13}{12}-1}$$

Subtract 1 from \frac{13}{12}.

$$\frac{13}{24}\sqrt[12]{z}$$

Evaluate

$$\frac{z^{\frac{13}{12}}}{2}$$