Solve 2^5z^-1/2^6z^-5

$$\frac { 2 ^ { 5 } z ^ { - 1 } } { 2 ^ { 6 } z ^ { - 5 } }$$

Evaluate

$$\frac{z^{4}}{2}$$

Solution Steps

Cancel out 2^{5} in both numerator and denominator.

$$\frac{\frac{1}{z}}{2z^{-5}}$$

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

$$\frac{z^{4}}{2}$$

Differentiate w.r.t. z

$$2z^{3}$$

Steps Using Definition of a Derivative

Cancel out 2^{5} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{\frac{1}{z}}{2z^{-5}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{4}}{2})$$

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

$$4\times \frac{1}{2}z^{4-1}$$

Multiply 4 times \frac{1}{2}.

$$2z^{4-1}$$

Subtract 1 from 4.

$$2z^{3}$$