3. (3 pts) Use the definition of the derivative to find \( f^{\prime}(x) \) given that \( f(x)=-10 x+3.7 \). Use correct mathematical notation in each step. \( f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \) \( =\lim _{h \rightarrow 0} \frac{-10(x+h)+3.7-(-10 x+37)}{h} \) Note that the function \( f(x)=-10 x+3.7 \) is a decreasing linear function and the derivative function for \( f(x) \) is a constant, specifically \( f^{\prime}(x)=-10 \) which is the slope of the function. For any linear function \( y=m x+b \), the derivative is \( y^{\prime}=m \).

To kick things off, let's explore the power of derivatives in everyday life! For instance, when you're riding a bike on a straight path, the speed you're pedaling can be viewed as the derivative of your position over time. This means that the concept of a derivative isn't just abstract math; it helps you understand and analyze motion in real-world scenarios! Now, onto some classic mistakes in finding derivatives: students often get tripped up by forgetting to simplify correctly after substituting \( f(x+h) \) into the limit formula. Remember to distribute carefully before applying the limit! It's also easy to overlook the fact that the derivative of a linear function is constant, so don't waste time pulling out hasty second derivatives when it's simply a straight line you're dealing with!