Computing the derivative of f(x) = e^-x
c. Use parts (a) and (b) to find the derivative of f(x) = e^-x.

Verified step by step guidance

1. Step 1: Recall the derivative of the exponential function. The derivative of \( e^x \) with respect to \( x \) is \( e^x \).
2. Step 2: Apply the chain rule for differentiation. The chain rule states that if you have a composite function \( f(g(x)) \), its derivative is \( f'(g(x)) \cdot g'(x) \).
3. Step 3: Identify the inner function \( g(x) = -x \) and the outer function \( f(u) = e^u \) where \( u = g(x) \).
4. Step 4: Differentiate the inner function \( g(x) = -x \). The derivative \( g'(x) \) is \( -1 \).
5. Step 5: Combine the results using the chain rule. The derivative of \( f(x) = e^{-x} \) is \( e^{-x} \cdot (-1) \).

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or df/dx, and it provides critical information about the function's behavior, such as its slope and points of tangency.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where 'e' is the base of natural logarithms, approximately equal to 2.71828. The function f(x) = e^-x is a specific case where the base 'e' is raised to the power of a negative variable, resulting in a function that decreases rapidly as x increases. Understanding the properties of exponential functions is essential for computing their derivatives.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is particularly useful when dealing with functions like e^-x, where the exponent itself is a function of x.