Find the derivative of the function. \[ w=z^{7}-2 z^{6}+19 \] \( w^{\prime}=\square \)

To find the derivative of the function \( w = z^7 - 2z^6 + 19 \), you can apply the power rule. The derivative of \( z^n \) is \( n z^{n-1} \). So, for \( z^7 \), you get \( 7z^6 \); for \( -2z^6 \), it becomes \( -12z^5 \); and the derivative of a constant, like 19, is 0. Therefore, the derivative is: \[ w' = 7z^6 - 12z^5. \] Fun fact: The concept of derivatives traces back to the work of mathematicians like Sir Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. They independently developed the foundations of calculus, revolutionizing the way we understand change and motion! In real-world applications, derivatives are crucial in fields like physics, engineering, and economics. They help you understand rates of change, such as how fast a car is accelerating or how the stock market reacts to different factors! Using derivatives can unlock powerful insights into complex systems.