Problem 42 - Calculus and its applications

Question

Differentiate. $$ f(x)=e^{-x^{2} / 2} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = -x e^{-x^2 / 2} \).

Step 1: Identify the Function Type

The given function, \( f(x) = e^{-x^2 / 2} \), is an exponential function where the exponent itself is a function of \( x \). It requires the use of the chain rule for differentiation since we have a composition of functions.

Step 2: Apply the Chain Rule

To differentiate \( f(x) = e^{-x^2 / 2} \), use the chain rule. The chain rule states that if you have a composite function \( e^{g(x)} \), the derivative is \( e^{g(x)} \times g'(x) \). Here, \( g(x) = -x^2 / 2 \).

Step 3: Find the Derivative of the Inner Function

Differentiate the inner function \( g(x) = -x^2 / 2 \). The derivative of \( -x^2 / 2 \) with respect to \( x \) is \( g'(x) = -x \), as the derivative of \( x^2 \) is \( 2x \) and the constant multiplier of \(-1/2\) remains.

Step 4: Combine Results to Find the Derivative

Using the chain rule, the derivative of \( f(x) = e^{-x^2 / 2} \) is:\[f'(x) = e^{-x^2 / 2} imes (-x) = -x imes e^{-x^2 / 2}\]This combines the derivative of the inner function \( g'(x) = -x \) with the original exponential function.

Key Concepts

Exponential Function

An exponential function is a mathematical concept where the variable appears as an exponent. This is different from polynomial functions, where the variable is the base. In the exponential function \( f(x) = e^{x} \), \( e \) is the base of the natural logarithm, approximately equal to 2.718. Exponential functions are used to model growth and decay processes due to their fundamental properties that represent constant relative change.

• The function \( f(x) = e^{-x^2/2} \) is a specific type of exponential function where the exponent is itself a function of \( x \).
• This makes it crucial to approach such problems with appropriate differentiation techniques like the chain rule.

Understanding exponential functions is key, as they are used not only in mathematics but also in fields like physics, engineering, and economics to model real-world phenomena.

Derivative of a Function

The derivative of a function represents the rate at which the function's value changes as its input changes. It provides valuable information about the behavior of a function at any point. For a function \( f(x) \), the derivative \( f'(x) \) is a new function that gives us the slope of the tangent line to the graph of \( f \) at any point.

• Calculating derivatives is a fundamental skill in calculus that allows us to determine instantaneous rates of change and understand how functions behave.
• For example, the derivative helps in finding the velocity if the original function represents position over time.

In our exercise, we find \( f'(x) = -x e^{-x^2/2} \). This derivative indicates how \( f(x) \) changes with respect to \( x \) and combines the behavior of both the exponential function and the influence of the exponent's rate of change.

Composite Function

A composite function is created when one function is applied to the results of another function. This scenario is common in calculus where chain rule application is necessary. If you have a function \( f(g(x)) \), \( f \) is applied to \( g(x) \), making it a composite function.

• The structure \( e^{-x^2/2} \) in the original problem is an example of a composite function.
• Here, \( g(x) = -x^2/2 \) and \( f(u) = e^{u} \), where \( u = g(x) \).

When differentiating composite functions, the chain rule is essential. It states that the derivative of \( f(g(x)) \) is \( f'(g(x)) \times g'(x) \). This approach allows us to unfold the nested structure of functions to understand how they interact, which is pivotal in solving calculus problems effectively.