Problem 7 - Thomas Calculus

Question

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. \begin{equation}y=e^{-5 x}\end{equation}

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = -5e^{-5x} \).

Step 1: Identify the Function to Differentiate

The function given is \( y = e^{-5x} \). We need to find its derivative with respect to \( x \).

Step 2: Apply the Exponential Derivative Rule

Recall that the derivative of \( e^{u} \) with respect to \( x \) is \( e^{u} \times \frac{du}{dx} \). Here, \( u = -5x \), so we need to find \( \frac{du}{dx} \).

Step 3: Differentiate the Exponent

Differentiate \( u = -5x \) with respect to \( x \). The derivative is \( \frac{du}{dx} = -5 \).

Step 4: Use the Chain Rule

Apply the chain rule to differentiate the function. The derivative of \( y = e^{-5x} \) is the original function \( e^{-5x} \) times the derivative of the exponent: \[ \frac{dy}{dx} = e^{-5x} \times (-5) \].

Step 5: Simplify the Derivative

Simplify the expression obtained from the differentiation: \( \frac{dy}{dx} = -5e^{-5x} \).

Key Concepts

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In the expression \( y = e^{-5x} \), the base \( e \) is the Euler's number, approximately equal to 2.71828. Such functions are powerful because they model growth and decay processes effectively, such as radioactive decay or population growth.

Exponential functions have a unique property: their rate of change is proportional to their value. This leads to their widespread use in real-world applications. In calculus, understanding how these functions change—i.e., finding their derivatives—is crucial.

Chain Rule

In calculus, the chain rule is a fundamental principle used to find the derivative of a composite function. A composite function is essentially a combination of two or more functions. The rule is essential when dealing with functions nested within each other, like \( y = e^{-5x} \).

Applying the chain rule involves two main steps:

• Differentiate the outer function, leaving the inner function unchanged.
• Multiply by the derivative of the inner function.

For our example of \( y = e^{-5x} \), the outer function is \( e^u \) with \( u = -5x \), and its derivative involves exponential and linear differentiation.

Differentiation

Differentiation is the process used to find the derivative of a function. It measures how a function changes as its input changes, and is akin to finding the slope of the tangent to the function's graph at any point.

In determining the derivative of \( y = e^{-5x} \), we used differentiation to find how \( y \) changes with respect to \( x \). This involves understanding how the rate of change of \( e^{-5x} \) depends on \( x \), which is particularly important in calculus for analyzing and interpreting graphs, economic models, and physical phenomena.

Exponential Derivative Rule

The exponential derivative rule provides a straightforward way to differentiate exponential functions. Specifically, for \( e^u \), the rule states that the derivative is \( e^u \times \frac{du}{dx} \). This is derived from the function's unique property where its derivative mirrors its original form.

In our example of \( y = e^{-5x} \), knowing \( u = -5x \), we apply this rule to find that the derivative is \( e^{-5x} \) multiplied by \( -5 \), resulting in \(-5e^{-5x}\). This rule is vital as it simplifies finding derivatives for complex exponential functions, emphasizing the elegance and simplicity in calculus.