Problem 38 - Calculus and its applications

Question

Differentiate. $$ f(x)=\frac{e^{x}}{x^{5}} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{e^x (x - 5)}{x^6} \).

Step 1: Recognize the Function Type

The function given is a quotient, which is of the form \( \frac{u}{v} \), where \( u = e^x \) and \( v = x^5 \). This tells us that we should use the quotient rule to differentiate the function.

Step 2: Recall the Quotient Rule

The quotient rule is given by \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \). We'll apply this to our function, where \( u = e^x \) and \( v = x^5 \).

Step 3: Calculate the Derivatives of u and v

Find the derivative of \( u \), which is \( u' = e^x \), since the derivative of \( e^x \) with respect to \( x \) is itself. Next, find the derivative of \( v \), which is \( v' = 5x^4 \), as the derivative of \( x^5 \) is \( 5x^4 \).

Step 4: Apply the Quotient Rule

Substitute \( u \), \( u' \), \( v \), and \( v' \) into the quotient rule formula: \[ f'(x) = \frac{e^x \cdot x^5 - e^x \cdot 5x^4}{(x^5)^2} \]

Step 5: Simplify the Expression

Simplify the expression by factoring out \( e^x x^4 \) from the numerator:\[ f'(x) = \frac{e^x x^4 (x - 5)}{x^{10}} \] Further simplification gives:\[ f'(x) = \frac{e^x (x - 5)}{x^6} \]

Key Concepts

Quotient Rule

When you encounter a function defined as a ratio of two other functions, it often calls for the quotient rule in differentiation. The quotient rule is a handy method for differentiating functions that take the form \( \frac{u}{v} \), where both \( u \) and \( v \) are functions of \( x \).

• **Formula**: The quotient rule states that \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \). This formula helps us find the derivative of the quotient directly.
• **How to Apply**: Begin by identifying \( u \) and \( v \) from your function. Next, differentiate both \( u \) and \( v \) independently to get \( u' \) and \( v' \). Penned these derivatives correctly in the formula is crucial to avoid sign errors.
• **Practical Tip**: Perform and simplify the calculations step by step. Simplifying at the end can make your work look cleaner and minimize the chance for arithmetic mistakes.

Applying this rule to our function \( f(x) = \frac{e^x}{x^5} \), helps us break it into manageable parts and correctly find its derivative.

Exponential Functions

Exponential functions may seem tricky at first glance, but they're among the most exciting in calculus due to their unique properties. An exponential function has a base constant raised to a variable power, most frequently \( e \), Euler's Number.- **Properties**: The key property of exponential functions like \( e^x \) is that they differentiate to themselves. This property simplifies the differentiation of these functions greatly because it remains constant. - **Importance in Differentiation**: When you differentiate \( e^x \), the result is just \( e^x \). Always remember to perform this step, especially when the function \( e^x \) appears in more complex expressions or in combination with other functions.In our exercise, knowing that \( u = e^x \) means that \( u' = e^x \). This property simplified the application of the quotient rule since one part of the differentiation was straightforward. This function remains integral in topics beyond basic differentiation, so mastering it early brings immense benefits.

Polynomial Derivatives

Differentiating polynomials is a fundamental skill in calculus, involving a simple technique called the power rule. The power rule states that for any polynomial \( x^n \), its derivative is \( nx^{n-1} \).- **Application**: This means the derivative of \( x^5 \) is \( 5x^4 \). You simply take the exponent as a coefficient and subtract one from the original exponent.- **Practical Use**: Apply this rule to every term in the polynomial separately if dealing with multi-term expressions. In more complex problems, combining roots, polynomial rules, and quotient rules is standard practice.Here, solving for \( v' = 5x^4 \) in our exercise gets us smoothly to apply the quotient rule by providing the necessary derivative factor. Despite its simplicity, fully grasping polynomial derivatives is essential, as these functions appear ubiquitously in calculus-based sciences.