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Exponential differentiation is a technique used to find the derivative of expressions that involve exponential functions. An exponential function can be expressed in the form \( e^{at} \), where \( e \) is the base of the natural logarithm and \( a \) is a constant. These functions grow or decay exponentially, depending on the value of \( a \). Clearly, differentiating these types of functions is key to understanding their behavior over time.

When differentiating an exponential function, there is a specific rule to follow. If you have a function \( y = e^{at} \), the derivative with respect to \( t \) is given by the formula \( \frac{dy}{dt} = ae^{at} \). In this, \( a \) is simply brought down as a multiplier, and the exponential function itself remains unchanged. This property makes exponential functions unique and their derivatives straightforward.

This rule is very useful when dealing with growth and decay problems, particularly in fields like biology, economics, and physics, where exponential growth or decay constants are frequently used.

The constant multiplier rule is a fundamental principle in differentiation that simplifies the process when there's a constant factor involved. When you take the derivative of a function where a constant multiplies an expression, this rule states that you can "factor out" the constant and then multiply it by the derivative of the remaining part of the expression. This makes it much easier to handle expressions with a constant component.

In the context of our example, \( B = 15e^{0.20t} \), 15 is the constant multiplier. According to the constant multiplier rule, you don't need to worry about differentiating this constant; just keep it as it is. You focus on differentiating the exponential part, which is \( e^{0.20t} \).

The rule can be represented by the formula: if \( y = c \cdot f(t) \), then the derivative \( \frac{dy}{dt} = c \cdot \frac{df(t)}{dt} \), where \( c \) is the constant. In our exercise, this means multiplying 15 by the derivative of \( e^{0.20t} \). This makes finding derivatives involving constants straightforward and manageable.

The derivative of exponential functions is a crucial concept in calculus, involving functions where the variable is in the exponent, like \( e^{at} \). These functions have unique derivatives due to the presence of the constant \( a \).

To differentiate an exponential function such as \( e^{at} \), you use the rule that says: the derivative with respect to \( t \) is \( ae^{at} \). This fact is what we applied in the original exercise to find \( \frac{dB}{dt} \).

In this exercise, we specifically applied these principles to the function \( B = 15e^{0.20t} \):

• First, recognize the part of the function where the exponential rule should be applied, which is \( e^{0.20t} \).
• Then, use the exponential differentiation rule to get \( 0.20e^{0.20t} \).
• Finally, apply the constant multiplier rule by multiplying 15 with the result from above, yielding \( 3e^{0.20t} \).

These steps highlight how working with exponential functions can be made simple using these well-defined rules.