Problem 22 - Calculus and its applications

Question

Differentiate. $$ f(x)=-3 e^{-x} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 3 e^{-x} \).

Step 1: Identify the Function Type

The function given is in the form of an exponential function, specifically of the form \[ f(x) = -3 e^{-x}. \]We need to differentiate this exponential function with respect to \( x \).

Step 2: Recall the Differentiation Rule for Exponential Functions

For the exponential function \( e^{g(x)} \), the derivative is \( e^{g(x)} \cdot g'(x) \). In our problem, we apply this rule to \( -3 e^{-x} \). The coefficient \(-3\) remains unchanged during differentiation.

Step 3: Differentiate the Exponent Function

Recognize the inner function in the exponent as \( g(x) = -x \). Now differentiate it: \( g'(x) = -1 \). In applying the differentiation rule, multiply the derivative of the exponent \( g'(x) \) by the original function.

Step 4: Apply the Chain Rule to Differentiate

Using the chain rule, differentiate \( f(x) = -3 e^{-x} \). Derivative of \( f(x) \) is: \[ f'(x) = -3 imes e^{-x} imes (-1) = 3 e^{-x}. \]

Step 5: Conclusion

Therefore, the derivative of the function is \[ f'(x) = 3 e^{-x}. \]

Key Concepts

Exponential Function

Exponential functions are a fundamental concept in calculus and appear frequently in both pure and applied mathematics. An exponential function has the form \( f(x) = a \cdot e^{g(x)} \), where \( e \) is the base of the natural logarithm (approximately 2.71828), and \( a \) is a constant. These functions are characterized by their rate of growth. The function \( e^{-x} \) is an exponential decay function due to the negative exponent, which indicates that as \( x \) increases, the function's value decreases.
Exponential functions are widely used in modeling real-world phenomena such as population growth, radioactive decay, and compound interest. Understanding their properties and behavior is crucial for applying calculus in practical scenarios.

Derivative Rules

When differentiating functions, knowing the appropriate rules is essential. Each function type has specific rules that apply to it. For exponential functions like \( f(x) = e^{x} \), the derivative is remarkably simple: it is the function itself, \( f'(x) = e^{x} \). However, in functions involving compositions, or those with coefficients or variable exponents, some additional steps are needed.
Here are some differentiation rules to keep in mind:

• The constant rule: The derivative of a constant is zero.
• The constant multiplication rule: If \( f(x) = c \cdot g(x) \), then \( f'(x) = c \cdot g'(x) \) where \( c \) is a constant.
• The exponential rule: For \( e^{g(x)} \), the derivative is \( e^{g(x)} \times g'(x) \).

It's these rules that help us tackle more complex differentiation problems, enabling us to break them down systematically.

Chain Rule

The chain rule is a powerful tool in calculus used to differentiate composite functions. A composite function is a function that is formed when one function is substituted into another. If you have a function \( f(x) = h(g(x)) \), the chain rule helps us differentiate it by taking the derivative of the outer function and multiplying it by the derivative of the inner function.
The formula for the chain rule is: \[ (f(g(x)))' = f'(g(x)) \cdot g'(x) \] In the context of the given exercise, the chain rule plays a crucial role. We have the function \( f(x) = -3 \cdot e^{-x} \), with the inner function \( g(x) = -x \). By applying the chain rule:

• Differentiate the outer function: \( e^{g(x)} \rightarrow e^{-x} \)
• Multiply by the derivative of the inner function \( g'(x) = -1 \)

Through these steps, the derivative of the composite function is determined, illustrating the chain rule’s utility in finding derivatives for functions with nested compositions.