Problem 15 - Applied Calculus

Question

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ y=t e^{-t^{2}} $$

Step-by-Step Solution

Answer

The derivative is \( y' = e^{-t^2} (1 - 2t^2) \).

Step 1: Identify the Derivative Rule

The given function is a product of two functions, \( y = t \cdot e^{-t^2} \). To find the derivative, we'll use the product rule, which is \( (uv)' = u'v + uv' \). In this case, \( u = t \) and \( v = e^{-t^2} \).

Step 2: Differentiate \( u = t \)

Differentiate \( u = t \) with respect to \( t \). The derivative is simple: \( u' = 1 \), because the derivative of \( t \) is 1.

Step 3: Differentiate \( v = e^{-t^2} \)

To differentiate \( v = e^{-t^2} \), use the chain rule. The outer function is \( e^x \) and the inner function is \( x = -t^2 \). First, find the derivative of \( e^x \), which is \( e^{x} \), and then multiply by the derivative of \( -t^2 \), which is \( -2t \). Thus, \( v' = e^{-t^2} \cdot (-2t) = -2te^{-t^2} \).

Step 4: Apply the Product Rule

Apply the product rule for derivatives: \( (uv)' = u'v + uv' \). Substitute the derivatives found:\[ y' = (1)(e^{-t^2}) + (t)(-2te^{-t^2}) \]

Step 5: Simplify the Expression

Simplify the expression derived from the product rule:\[ y' = e^{-t^2} - 2t^2e^{-t^2} \]. Factor out \( e^{-t^2} \) if needed:\[ y' = e^{-t^2} (1 - 2t^2) \].

Key Concepts

Product Rule

The product rule is an essential tool in calculus, especially when dealing with the derivatives of functions that are multiplied together. Imagine you have two functions, say \( u(t) \) and \( v(t) \). If these functions are multiplied to form a new function \( y(t) = u(t) \cdot v(t) \), we need to use the product rule to find the derivative of \( y(t) \).
The product rule states:

• If \( y = uv \), then the derivative \( y' = u'v + uv' \).

This means you take the derivative of the first function (\( u' \)), multiply it by the second function (\( v \)), then add the derivative of the second function (\( v' \)), multiplied by the first function (\( u \)).
Applying the product rule can simplify complex derivatives, allowing us to handle functions expressed as the product of two simpler functions.

Chain Rule

The chain rule is another fundamental concept in calculus, used to differentiate composite functions. Composite functions are functions composed of other functions, like \( v(t) = e^{-t^2} \), where the exponent involves a function of \( t \).
Here's how the chain rule works:

• If a function \( y = f(g(t)) \), where \( f \) is a function of \( g \) which in turn is a function of \( t \), then the derivative is found using \( y' = f'(g(t)) \cdot g'(t) \).

This rule helps us break down the differentiation process into manageable steps. First, we differentiate the outer function, treating the inner function as a separate variable. Then, multiply this by the derivative of the inner function itself.
The chain rule is powerful because it lets you handle functions nested within other functions, making complex derivatives more approachable.

Differentiation

Differentiation is the mathematical process of finding derivatives, which are the rates at which quantities change. In simple terms, a derivative represents how a function's output changes as its input changes.
Differentiation is vital in calculus and has many applications in physics, engineering, economics, and beyond.

• Basic differentiation rules include the constant rule, power rule, and sum rule.
• Advanced rules like the product rule and chain rule help handle more complex functions.

Differentiation can be thought of as finding the "slope" of the function graph at any given point, giving insights into the function's behavior. Calculating derivatives often leads to understanding how systems evolve and assisting in optimizing processes.

Exponential Function

Exponential functions are a unique class of functions characterized by their constant relative rate of growth. A typical exponential function looks like \( f(t) = e^t \), where \( e \) is the base of natural logarithms, approximately 2.718.
Critical features of exponential functions include:

• Their derivatives are proportional to the function itself, i.e., the derivative of \( e^t \) is also \( e^t \).
• Exponential functions model a wide range of real-world processes, such as growth and decay.

When the exponent is not just a simple variable but another function, as in \( e^{-t^2} \), the chain rule becomes important for differentiation. In this context, the exponential function's self-replicating derivative property simplifies calculations while maintaining the essence of the function's behavior. Understanding exponential functions provides powerful insights into natural systems and mathematical modeling.