Solve z/z^2*z^86*z^-44

$$\frac { z } { z ^ { 2 } \cdot z ^ { 86 } \cdot z ^ { - 44 } }$$

Evaluate

$$\frac{1}{z^{43}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 86 to get 88.

$$\frac{z}{z^{88}z^{-44}}$$

To multiply powers of the same base, add their exponents. Add 88 and -44 to get 44.

$$\frac{z}{z^{44}}$$

Rewrite z^{44} as zz^{43}. Cancel out z in both numerator and denominator.

$$\frac{1}{z^{43}}$$

Differentiate w.r.t. z

$$-\frac{43}{z^{44}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{44}}{z^{86}}z^{1-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{1}{z^{42}}\times \frac{1}{z})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-\frac{1}{z^{42}}z^{-1-1}$$

Do the arithmetic.

$$\left(-\frac{1}{z^{42}}\right)z^{-2}$$