Problem 5 - Applied Calculus

Question

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ y=5 x e^{x^{2}} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = e^{x^2}(5 + 10x^2) \).

Step 1: Identify the Function

First, recognize that the given function is in the form of a product, specifically \( y = 5x \cdot e^{x^2} \). Here, \( u = 5x \) and \( v = e^{x^2} \). We will need to use the product rule to differentiate.

Step 2: Recall the Product Rule

To differentiate a product of two functions \( u \cdot v \), use the product rule: \( \frac{d}{dx}(u \cdot v) = u'v + uv' \). We will apply this rule to find \( \frac{dy}{dx} \).

Step 3: Differentiate Each Component

Differentiate \( u = 5x \) to get \( u' = 5 \). Differentiate \( v = e^{x^2} \) using the chain rule: \( v' = \frac{d}{dx}(e^{x^2}) = 2xe^{x^2} \).

Step 4: Apply the Product Rule

Substitute \( u' \), \( u \), \( v \), and \( v' \) into the product rule formula: \( \frac{dy}{dx} = 5 \cdot e^{x^2} + 5x \cdot 2xe^{x^2} \).

Step 5: Simplify the Expression

Simplify the derivative: \( \frac{dy}{dx} = 5e^{x^2} + 10x^2e^{x^2} \). Factor out \( e^{x^2} \) from the expression to get \( \frac{dy}{dx} = e^{x^2}(5 + 10x^2) \).

Key Concepts

Product Rule

The product rule is an essential technique in calculus used to differentiate functions that are products of two simpler functions. With the function from our exercise, which is given by

• \( y = 5x \cdot e^{x^2} \)

we see it's a product of the functions \( u = 5x \) and \( v = e^{x^2} \).
To differentiate this, the product rule formula states:

• \[ \frac{d}{dx}(uv) = u'v + uv' \]

Here, \( u' \) and \( v' \) denote the derivatives of \( u \) and \( v \) respectively. By substituting the derivatives of \( u \) and \( v \) into this formula, we can effectively find the derivative of the entire product. This makes the product rule a powerful tool whenever dealing with two functions multiplied together.

Chain Rule

The chain rule is another pivotal rule in calculus, used for differentiating composite functions. In the derivative calculation of \( e^{x^2} \), the chain rule is applied. The expression \( e^{x^2} \) can be seen as a composite function, namely \( e^{u} \) where \( u = x^2 \).
When differentiating \( e^{x^2} \), the chain rule allows us to break down the process into more manageable parts:

• First, differentiate the outer function: the derivative of \( e^u \) with respect to \( u \) is \( e^u \).
• Then, differentiate the inside function: the derivative of \( x^2 \) with respect to \( x \) is \( 2x \).

Multiply these results to get the derivative of the composite function:\[v'(x) = 2xe^{x^2}\]The chain rule hence simplifies complex differentiations by dissecting them into easier steps.

Derivative Calculation

In calculus, calculating the derivative of a function is crucial as it represents the rate of change. In the original exercise, we were tasked with finding the derivative of

• \( y = 5x \cdot e^{x^2} \)

We use both the product rule and the chain rule together for an efficient calculation process.
Let's break it down further:

• Differentiate \( u = 5x \) to get \( u' = 5 \).
• Apply the chain rule to \( v = e^{x^2} \) to find \( v' = 2xe^{x^2} \).

With these derivatives, substitute into the product rule formula to achieve:

• \( \frac{dy}{dx} = u'v + uv' = 5e^{x^2} + 5x \cdot 2xe^{x^2} \)

After simplifying, we get:

• \( \frac{dy}{dx} = e^{x^2}(5 + 10x^2) \)

This derivative gives us the rate of change of \( y \) with respect to \( x \). The process demonstrates the power of calculus in employing rules to handle complex expressions.

Exponential Functions

Exponential functions are a key element in calculus and other areas of mathematics. They are of the form \( y = a^x \) or the natural exponential function \( y = e^x \), which is encountered in the original problem.
The natural exponential function, \( e^{x^2} \), exhibits unique properties:

• The base \( e \) is approximately equal to 2.71828, which has special properties in calculus.
• Derivative of \( e^x \) with respect to \( x \) is \( e^x \), making it easy to manage compared to other functions.

When dealing with more complex exponential functions like \( e^{x^2} \), the chain rule is often employed, as its composite nature requires deeper analysis. Understanding these functions is vital, as they frequently occur in natural processes, economics, and many scientific calculations.