Solve (z^2)^2/(z^6)^8=

$$\frac { ( z ^ { 2 } ) ^ { 2 } } { ( z ^ { 6 } ) ^ { 8 } } =$$

Evaluate

$$\frac{1}{z^{44}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{z^{4}}{\left(z^{6}\right)^{8}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 8 to get 48.

$$\frac{z^{4}}{z^{48}}$$

Rewrite z^{48} as z^{4}z^{44}. Cancel out z^{4} in both numerator and denominator.

$$\frac{1}{z^{44}}$$

Differentiate w.r.t. z

$$-\frac{44}{z^{45}}$$

Steps Using Chain Rule

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{4}}{\left(z^{6}\right)^{8}})$$

To raise a power to another power, multiply the exponents. Multiply 6 and 8 to get 48.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{z^{4}}{z^{48}})$$

Rewrite z^{48} as z^{4}z^{44}. Cancel out z^{4} in both numerator and denominator.

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

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).

$$-\left(z^{44}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}z}(z^{44})$$

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}.

$$-\left(z^{44}\right)^{-2}\times 44z^{44-1}$$

Simplify.

$$-44z^{43}\left(z^{44}\right)^{-2}$$