Solve frac{(z^{3})^{-4}*(z^{-5})^{-4}(z^{3})^{-2}}{(z^{-3})^{-5}(z^-5)^0(z)(z^-3)^-2}

$$\frac { ( z ^ { 3 } ) ^ { - 4 } \cdot ( z ^ { - 5 } ) ^ { - 4 } ( z ^ { 3 } ) ^ { - 2 } } { ( z ^ { - 3 } ) ^ { - 5 } ( z ^ { - 5 } ) ^ { 0 } ( z ) ( z ^ { - 3 } ) ^ { - 2 } }$$

Evaluate

$$\frac{1}{z^{20}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 3 and -4 to get -12.

$$\frac{z^{-12}\left(z^{-5}\right)^{-4}\left(z^{3}\right)^{-2}}{\left(z^{-3}\right)^{-5}\left(z^{-5}\right)^{0}z\left(z^{-3}\right)^{-2}}$$

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

$$\frac{z^{-12}z^{20}\left(z^{3}\right)^{-2}}{\left(z^{-3}\right)^{-5}\left(z^{-5}\right)^{0}z\left(z^{-3}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -2 to get -6.

$$\frac{z^{-12}z^{20}z^{-6}}{\left(z^{-3}\right)^{-5}\left(z^{-5}\right)^{0}z\left(z^{-3}\right)^{-2}}$$

To multiply powers of the same base, add their exponents. Add -12 and 20 to get 8.

$$\frac{z^{8}z^{-6}}{\left(z^{-3}\right)^{-5}\left(z^{-5}\right)^{0}z\left(z^{-3}\right)^{-2}}$$

To multiply powers of the same base, add their exponents. Add 8 and -6 to get 2.

$$\frac{z^{2}}{\left(z^{-3}\right)^{-5}\left(z^{-5}\right)^{0}z\left(z^{-3}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -5 to get 15.

$$\frac{z^{2}}{z^{15}\left(z^{-5}\right)^{0}z\left(z^{-3}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply -5 and 0 to get 0.

$$\frac{z^{2}}{z^{15}z^{0}z\left(z^{-3}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -2 to get 6.

$$\frac{z^{2}}{z^{15}z^{0}zz^{6}}$$

To multiply powers of the same base, add their exponents. Add 15 and 0 to get 15.

$$\frac{z^{2}}{z^{15}zz^{6}}$$

To multiply powers of the same base, add their exponents. Add 15 and 1 to get 16.

$$\frac{z^{2}}{z^{16}z^{6}}$$

To multiply powers of the same base, add their exponents. Add 16 and 6 to get 22.

$$\frac{z^{2}}{z^{22}}$$

Rewrite z^{22} as z^{2}z^{20}. Cancel out z^{2} in both numerator and denominator.

$$\frac{1}{z^{20}}$$

Differentiate w.r.t. z

$$-\frac{20}{z^{21}}$$

Steps Using Chain Rule

To raise a power to another power, multiply the exponents. Multiply 3 and -4 to get -12.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{-12}\left(z^{-5}\right)^{-4}\left(z^{3}\right)^{-2}}{\left(z^{-3}\right)^{-5}\left(z^{-5}\right)^{0}z\left(z^{-3}\right)^{-2}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{-12}z^{20}\left(z^{3}\right)^{-2}}{\left(z^{-3}\right)^{-5}\left(z^{-5}\right)^{0}z\left(z^{-3}\right)^{-2}})$$

To raise a power to another power, multiply the exponents. Multiply 3 and -2 to get -6.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{-12}z^{20}z^{-6}}{\left(z^{-3}\right)^{-5}\left(z^{-5}\right)^{0}z\left(z^{-3}\right)^{-2}})$$

To multiply powers of the same base, add their exponents. Add -12 and 20 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{8}z^{-6}}{\left(z^{-3}\right)^{-5}\left(z^{-5}\right)^{0}z\left(z^{-3}\right)^{-2}})$$

To multiply powers of the same base, add their exponents. Add 8 and -6 to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{2}}{\left(z^{-3}\right)^{-5}\left(z^{-5}\right)^{0}z\left(z^{-3}\right)^{-2}})$$

To raise a power to another power, multiply the exponents. Multiply -3 and -5 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{2}}{z^{15}\left(z^{-5}\right)^{0}z\left(z^{-3}\right)^{-2}})$$

To raise a power to another power, multiply the exponents. Multiply -5 and 0 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{2}}{z^{15}z^{0}z\left(z^{-3}\right)^{-2}})$$

To raise a power to another power, multiply the exponents. Multiply -3 and -2 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{2}}{z^{15}z^{0}zz^{6}})$$

To multiply powers of the same base, add their exponents. Add 15 and 0 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{2}}{z^{15}zz^{6}})$$

To multiply powers of the same base, add their exponents. Add 15 and 1 to get 16.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{2}}{z^{16}z^{6}})$$

To multiply powers of the same base, add their exponents. Add 16 and 6 to get 22.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{2}}{z^{22}})$$

Rewrite z^{22} as z^{2}z^{20}. Cancel out z^{2} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{1}{z^{20}})$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\left(z^{20}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}z}(z^{20})$$

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}.

$$-\left(z^{20}\right)^{-2}\times 20z^{20-1}$$

Simplify.

$$-20z^{19}\left(z^{20}\right)^{-2}$$