Problem 49 - Calculus and its applications

Question

Differentiate. $$ y=1-e^{-3 x} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Function

The given function is a combination of constant terms and an exponential function. Specifically, it is of the form \( y = 1 - e^{-3x} \).

Step 2: Differentiate Constant Terms

Differentiate the constant term \( 1 \). The derivative of a constant is zero, so the derivative of \( 1 \) is \( 0 \).

Step 3: Apply Exponential Differentiation

Now, differentiate the term \( -e^{-3x} \). Recall that the derivative of \( e^{u} \) with respect to \( x \), where \( u \) is a function of \( x \), is \( e^{u} \cdot \frac{du}{dx} \).

Step 4: Differentiate the Exponential Argument

Here, \( u = -3x \), so \( \frac{du}{dx} = -3 \).

Step 5: Compute the Derivative

Using the chain rule, the derivative of \( -e^{-3x} \) is \( -e^{-3x} \cdot (-3) = 3e^{-3x} \).

Step 6: Combine the Results

Combine the results of the differentiated components: derivative of constant is zero, and derivative of \( -e^{-3x} \) is \( 3e^{-3x} \). Thus, the overall derivative of the function is \( y' = 0 + 3e^{-3x} = 3e^{-3x} \).

Key Concepts

Exponential Function

The exponential function is one of the most important functions in mathematics, especially when dealing with continuous growth or decay. An exponential function is typically written as \( e^{x} \), where \( e \) is approximately equal to 2.71828 - a constant known as Euler's number. This function has unique properties, such as the rate of change being proportionate to the function's current value.

An important feature of exponential functions is that they grow or decay at a constant rate. They are incredibly useful in modeling various real-world scenarios, such as population growth, radioactive decay, and continuously compounded interest. In differentiation, recognizing the form of an exponential function is crucial, as it dictates the rules and steps needed to find derivatives effectively.

Chain Rule

The chain rule is a fundamental technique in calculus for finding derivatives of composite functions. A composite function is formed when one function is applied to the result of another function, like \( f(g(x)) \).

To apply the chain rule, follow these steps:

• Differentiate the outer function while keeping the inner function unchanged.
• Multiply the result by the derivative of the inner function.

The rule can be expressed mathematically as:\[ \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \]In the original problem, the chain rule is used to differentiate the function \( -e^{-3x} \). First, the derivative of \( e^{u} \) is taken, resulting in \( e^{u} \cdot \frac{du}{dx} \), where \( u = -3x \) and \( \frac{du}{dx} = -3 \). This process converts complex differentiation into manageable pieces.

Derivative of a Constant

The derivative of a constant is one of the simplest yet foundational ideas in calculus. A constant is a number without any variable attached, like 1, \( \pi \), or -7. When you differentiate any constant term, the result is always zero because constants do not change. Thus, their rate of change is zero.

For example, if you have a function \( y = 5 \) or in the original exercise, \( y = 1 \), differentiating it yields 0. This principle is extremely useful as it simplifies differentiation by eliminating terms that do not depend on the variable in question, allowing focus on those that do. Recognizing and applying the derivative of a constant efficiently is essential in quickly solving more complex calculus problems.