Problem 33 - Calculus and its applications

Question

Differentiate. $$ F(x)=\frac{e^{2 x}}{x^{4}} $$

Step-by-Step Solution

Answer

The derivative is \( F'(x) = \frac{2e^{2x} (x - 2)}{x^5} \).

Step 1: Identify the Rule to Use

The given function, \( F(x) = \frac{e^{2x}}{x^{4}} \), can be differentiated using the Quotient Rule. The Quotient Rule states that if you have a function \( \frac{u}{v} \), its derivative is \( \frac{v \cdot u' - u \cdot v'}{v^2} \). Here, \( u = e^{2x} \) and \( v = x^4 \).

Step 2: Differentiate the Numerator (u)

Determine \( u' \), the derivative of the numerator. Since \( u = e^{2x} \), we use the chain rule to get \( u' = 2e^{2x} \) because the derivative of \( e^{2x} \) is \( 2e^{2x} \).

Step 3: Differentiate the Denominator (v)

Find \( v' \), the derivative of the denominator. For \( v = x^4 \), the derivative is \( v' = 4x^3 \) using the power rule.

Step 4: Apply the Quotient Rule

Substitute \( u \), \( u' \), \( v \), and \( v' \) into the Quotient Rule: \[ F'(x) = \frac{x^4\cdot 2e^{2x} - e^{2x}\cdot 4x^3}{x^8} \] Simplify the expression, noting the common factor of \( e^{2x} \): \[ F'(x) = \frac{e^{2x} [2x^4 - 4x^3]}{x^8} \]

Step 5: Simplify the Expression

Factor out common terms. In \( 2x^4 - 4x^3 \), factor out \( 2x^3 \):\[ F'(x) = \frac{e^{2x} \cdot 2x^3 (x - 2)}{x^8} \] Simplify the fraction by canceling \( x^3 \) from both numerator and denominator:\[ F'(x) = \frac{2e^{2x} (x - 2)}{x^5} \]

Step 6: Final Simplified Derivative

The derivative of the function \( F(x) = \frac{e^{2x}}{x^4} \) is:\[ F'(x) = \frac{2e^{2x} (x - 2)}{x^5} \]

Key Concepts

Chain Rule

The Chain Rule is a vital tool in calculus used to find the derivative of composite functions. Imagine you're peeling the layers of an onion; each layer represents a function inside another. To differentiate a composite function, you have to follow the rule of peeling off one layer at a time.

Consider a function \( f(g(x)) \). To find its derivative, the Chain Rule states that you differentiate the outer function and multiply it by the derivative of the inner function. This can be expressed as \( f'(g(x)) \cdot g'(x) \).

• Find the derivative of the outer function, treating the inner function as a variable.
• Multiply this by the derivative of the inner function.

In the context of the function \( e^{2x} \), the outer function is the exponential function \( e^u \), and the inner function is \( u = 2x \). This results in the derivative \( 2e^{2x} \) when applying the Chain Rule.

Making use of the Chain Rule allows us to efficiently compute derivatives of complex, nested functions by focusing on each layer individually.

Power Rule

The Power Rule is one of the most straightforward differentiation rules in calculus. It helps in finding the derivative of polynomial functions, where the function is in the form \( x^n \). The rule simplifies to the derivative as follows: bring the power down as a coefficient and decrease the power by one. This is expressed mathematically as \( \frac{d}{dx}x^n = nx^{n-1} \).

Using the Power Rule efficiently simplifies the differentiation process for polynomial expressions. In the given exercise, the denominator \( x^4 \) was differentiated to become \( 4x^3 \). By applying the Power Rule:

• Recognize the base \( x \) as the variable being affected.
• Drop the exponent and reduce it by one.
• So, from \( x^4 \), we obtain \( 4x^3 \).

The Power Rule makes differentiation quick and easy, especially for polynomials, which often appear in calculus problems.

Derivative Simplification

After computing the derivative using differentiation rules, it’s crucial to simplify the expression. This helps in understanding the derivative better and makes the final expression more manageable.

In our exercise, once the Quotient Rule produced the derivative \( F'(x) = \frac{x^4\cdot 2e^{2x} - e^{2x}\cdot 4x^3}{x^8} \), simplification was needed. Here are the steps that can aid in simplifying derivatives:

• Identify and factor out common elements in the numerator and denominator. In this case, \( e^{2x} \) was common.
• Factor terms in the numerator such as \( 2x^3 \) in \( 2x^4 - 4x^3 \).
• Cancel common factors across numerator and denominator. Here, \( x^3 \) was canceled.

The result is a cleaner and more concise expression \( \frac{2e^{2x} (x - 2)}{x^5} \).

Such simplification steps not only refine the derivative but also prepare it for further analysis or integration as needed.