Find the derivative of the function. \[ w=z^{7}-3 z^{6}+12 \] \( w^{\prime}=\square \)

To find the derivative of the function \( w = z^{7} - 3z^{6} + 12 \), we apply the power rule. The derivative of \( z^n \) is \( nz^{n-1} \). Thus, for the given function: 1. The derivative of \( z^{7} \) is \( 7z^{6} \). 2. The derivative of \( -3z^{6} \) is \( -18z^{5} \). 3. The derivative of the constant \( 12 \) is \( 0 \). Putting this all together, we get: \[ w' = 7z^{6} - 18z^{5} \] Now, filling in the blank, we have \( w' = 7z^{6} - 18z^{5} \). This function exhibits beautiful behavior; it can be used to describe various phenomena in physics and engineering like growth rates, decay processes, and more! The powers and constants give it a flavor of predictability and dynamics combined. Plus, as we engage with this polynomial, we develop our ability to effortlessly maneuver through calculus concepts like higher derivatives, which can lead us into graphing and optimization problems!