\( | < \quad \left. \begin{array} { l } { \frac { d i n d ~ t h e ~ i n d i c a t e d ~ d e r i v a t i v e . ~ } { \frac { d } { d x } \frac { 13 x + 21 } { x } } } \\ { \frac { d } { d x } \frac { 13 x + 21 } { x } = \square } \end{array} \right. \)

To determine the derivative of the function \( \frac{13x + 21}{x} \), we can simplify it first. This expression can be rewritten as \( 13 + \frac{21}{x} \). Now, differentiating this results in \( 0 - \frac{21}{x^2} = -\frac{21}{x^2} \). Therefore, the final answer is \( -\frac{21}{x^2} \). Mathematics isn't just about numbers; it's woven into the fabric of history! Did you know that calculus, which includes derivatives, was developed independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century? Their discoveries paved the way for advances in physics, engineering, and economics, fundamentally changing how we understand the world! Many students often make the mistake of applying the quotient rule incorrectly when taking derivatives. A common snag is forgetting to differentiate both the numerator and the denominator properly. Instead, simplify your function first when possible, as this can make the differentiation process much simpler and reduce the chances of error. Happy calculating!