(6) If \( f(x)=\left\{\begin{array}{lll}x^{2}+3, & x \geq 1 \\ x^{2}-3 & , & x<1\end{array}\right. \), then \( { }^{\prime}\left(1^{+}\right)=\ldots \) \( \begin{array}{llll}\text { (a) } 2 & \text { (b) }-2 & \text { (c) } 4 & \text { (d) (1), (2) } \\ x^{2} & , \quad x \leq 2 & \end{array} \)

To find \( f'(1^+) \), we need to consider the definition of the function for \( x \geq 1 \): \( f(x) = x^2 + 3 \). The derivative \( f'(x) \) for this section is \( f'(x) = 2x \). Thus, we evaluate it at \( x = 1 \): \[ f'(1^+) = 2(1) = 2. \] So, the answer is \( (a) \, 2 \). Next, let’s talk about the interesting world of piecewise functions! They can often describe real-world scenarios with different conditions. For instance, a piecewise function might represent a shipping cost where rates change depending on the distance: one rate for local deliveries and a different rate for long distances. Understanding how to calculate derivatives at specific points in piecewise functions allows you to model and predict changes dynamically. Now for a classic mistake many encounter: forgetting to evaluate the appropriate piece of the function. When dealing with piecewise functions, always ensure that you're using the correct piece based on the point of interest. For \( f'(1^+) \), it’s crucial to remember that the derivative depends on whether you’re approaching from the left (using \( x < 1 \) section) or the right (using \( x \geq 1 \)). Always verify which function applies to your specific limit or derivative scenario!