91Ó°ÊÓ

In calculus, the concept of the derivative is central to understanding how functions change. At a basic level, a derivative represents the rate at which a function's value changes with respect to a change in its input (usually denoted as the variable \( x \)).
Think of it as measuring the function's sensitivity to change. It's like asking, "If I tweak \( x \) just a little, how much will \( f(x) \) change?".

• The derivative gives us important information about the function's behavior, including where it increases or decreases.
• It's particularly useful for finding the slope of a function at any given point.

To compute the derivative of a function like \( f(x) = 2 + xe^x \), we perform differentiation. This mathematical process involves applying various rules to find how functions change at every point. For example, the term \( 2 \) is constant, so its derivative is \( 0 \). But for more complex parts, like \( xe^x \), we employ specific techniques such as the product rule to handle the interplay of different variables.

The product rule is a crucial tool in calculus used to differentiate functions that are the product of two or more separate functions. When dealing with composite functions such as \( xe^x \), where you have one function (\( x \)) multiplied by another (\( e^x \)), we rely on the product rule.
The rule is simple but powerful: it states that the derivative of a product of two functions \( u(x) \) and \( v(x) \) is given by \((uv)' = u'v + uv'\). To use this, you need to:

• Differentiate each function separately. For \( u(x) = x \), we get \( u'(x) = 1 \).
• Differentiate \( v(x) = e^x \), which gives \( v'(x) = e^x \).
• Then, substitute these into the product rule formula: \( f'(x) = 1 \cdot e^x + x \cdot e^x \).

In our original exercise, applying the product rule to find the derivative of \( xe^x \) results in \( f'(x) = e^x(1+x) \). This provides us with the rate of change of \( f \) with respect to \( x \), which is crucial for finding slopes.

Finding the slope of a curve at a given point is a common problem in calculus. The concept of slope is familiar from basic algebra, where it measures steepness. In calculus, the derivative of a function at a point tells us the slope of the tangent line to the graph of the function at that point.
To determine the slope, we first find the derivative of the function, which represents a general formula for the slope of the tangent at any point \( x \).

• Once we have the derivative, like \( f'(x) = e^x(1+x) \), we substitute the given \( x \)-coordinate into it to find the slope at that particular point.
• In the example \( f(x)=2+xe^x \), evaluating the derivative at \( x=0 \) gives \( f'(0) = 1 \), indicating the slope at the point \((0,2)\) is 1.

Understanding how the derivative leads to the slope of the graph can help with visual interpretations of mathematical functions. The slope illustrates how steep the graph is at any given point, which is essential for predicting and analyzing change.