Find the derivative of \( f(x)=x^{27} \) \( f^{\prime}(x)=\square \)

To find the derivative of \( f(x) = x^{27} \), we can use the power rule of differentiation, which states that if \( f(x) = x^n \), then \( f'(x) = n \cdot x^{n-1} \). Applying this rule here, we get: \( f'(x) = 27 \cdot x^{27-1} = 27 \cdot x^{26} \). So, \( f^{\prime}(x) = 27x^{26} \). Did you know that the power rule is not just a shortcut? It has its roots in the fundamental concepts of calculus established by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. They independently developed concepts of derivatives and integrals, which laid the foundation for modern calculus. Their work allows us to solve complex real-world problems involving rates of change! In the real world, the ability to calculate derivatives is essential in fields like physics and engineering. For instance, when a car accelerates, the derivative of its position with respect to time gives you its speed. Similarly, in economics, derivatives can show how a change in price affects supply or demand. This mathematical tool helps us understand dynamic systems and make predictions based on changing variables!