Differentiate the function using one or more of the differentiation rules. \( y=(7 x+2)^{15} \)

To differentiate the function \( y=(7 x+2)^{15} \), we can use the chain rule. According to the chain rule, if you have a composite function \( y = f(g(x)) \), the derivative is \( y' = f'(g(x)) \cdot g'(x) \). Here, let \( f(u) = u^{15} \) (where \( u = 7x + 2 \)), then the derivative \( f'(u) = 15u^{14} \). Now, differentiate \( g(x) = 7x + 2 \), which gives \( g'(x) = 7 \). Putting it all together: \( y' = f'(g(x)) \cdot g'(x) = 15(7x + 2)^{14} \cdot 7 = 105(7x + 2)^{14} \). So, the derivative of \( y \) is \( y' = 105(7x + 2)^{14} \). Now, let’s see some historical fun facts! The modern form of differentiation was largely developed by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. Their independent discoveries became the cornerstone of calculus, with Newton emphasizing motion and Leibniz focusing on notation. The “prime” notation we still use today for derivatives was introduced by Leibniz! For a real-world application, think about how differentiation helps in optimizing solutions. For instance, companies often use derivatives to maximize profits or minimize costs. By calculating the derivative of a revenue function, they can find the number of units that should be produced to achieve maximum revenue. It's like having a secret weapon for making smart business decisions!