Problem 22 - Calculus - AP Edition

Question

Find \(d y / d x\). $$y=\frac{e^{x}}{\ln x}$$

Step-by-Step Solution

Answer

The derivative is \( \frac{e^x (\ln x - \frac{1}{x})}{(\ln x)^2} \).

Step 1: Identify Differentiation Rule

The function given is a quotient, \( y = \frac{e^x}{\ln x} \). We need to apply the Quotient Rule for derivatives, which states that if \( y = \frac{u}{v} \), then \( \frac{d y}{d x} = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^2} \), where \( u = e^x \) and \( v = \ln x \).

Step 2: Differentiate the Numerator

Differentiate the numerator \( u = e^x \). The derivative of \( e^x \) with respect to \( x \) is \( \frac{d u}{d x} = e^x \).

Step 3: Differentiate the Denominator

Differentiate the denominator \( v = \ln x \). The derivative of \( \ln x \) with respect to \( x \) is \( \frac{d v}{d x} = \frac{1}{x} \).

Step 4: Apply the Quotient Rule

Substitute the derivatives found in Steps 2 and 3 into the quotient rule formula: \[ \frac{d y}{d x} = \frac{(\ln x)(e^x) - (e^x)(\frac{1}{x})}{(\ln x)^2} \].

Step 5: Simplify the Expression

Simplify the expression obtained in Step 4: \( \frac{d y}{d x} = \frac{e^x \ln x - \frac{e^x}{x}}{(\ln x)^2} \). Factoring out \( e^x \) from the numerator gives \( \frac{d y}{d x} = \frac{e^x (\ln x - \frac{1}{x})}{(\ln x)^2} \). This is the final simplified derivative.

Key Concepts

Quotient Rule

In calculus, the Quotient Rule is a fundamental tool for differentiating functions that are expressed as one function divided by another. Specifically, if you have a function of the form \( y = \frac{u}{v} \), where both \( u \) and \( v \) are functions of \( x \), the rule provides a way to find the derivative, \( \frac{d y}{d x} \). The formula is:

• \( \frac{d y}{d x} = \frac{v \cdot \frac{d u}{d x} - u \cdot \frac{d v}{d x}}{v^2} \)

To remember it easily, think of it as "low \( \cdot \) derivative of high minus high \( \cdot \) derivative of low, divided by low squared." This statement captures the essence of multiplying the bottom function by the derivative of the top, subtracting the top function times the derivative of the bottom, and then dividing everything by the square of the bottom function. It is crucial in scenarios such as this problem with \( y = \frac{e^x}{\ln x} \).

Exponential Functions

Exponential functions are functions where a constant base is raised to a variable exponent, such as \( e^x \). Here, \( e \) is an important mathematical constant approximately equal to 2.718. The beauty of exponential functions, particularly \( e^x \), lies in their differentiability. The derivative of \( e^x \) with respect to \( x \) is very intuitive and nicely self-replicating:

• \( \frac{d}{d x} e^x = e^x \)

This unique property makes \( e^x \) a powerful component in calculus as it simplifies differentiation. In this context, exponential functions also show us how growth processes behave, given \( e \) is often the base in natural exponential processes. For our function \( y = \frac{e^x}{\ln x} \), it provided the top function in our quotient.

Logarithmic Differentiation

Logarithmic differentiation is a technique used when differentiating complex functions involving logarithms. Typically, it helps simplify the differentiation process, especially when the function is a product or a quotient involving variables raised to powers. The basic idea is to take the natural log of both sides of a function \( y = f(x) \), then differentiate using implicit differentiation.In the exercise, we differentiate \( \ln x \), leading to:

• The derivative is \( \frac{1}{x} \)

Logarithmic derivatives are particularly streamlined thanks to the derivative property of natural logs. In this problem, the denominator \( \ln x \), made it necessary to recognize and apply logarithmic differentiation as part of the overall Quotient Rule application.

Chain Rule

Though not directly used in the direct application provided in this problem, the Chain Rule is another differentiation technique often paired with functions involving exponential and logarithmic forms. It is used when one or more functions are nested within another, like \( f(g(x)) \). The rule states that:

• \( \frac{d}{d x} f(g(x)) = f'(g(x)) \cdot g'(x) \)

This means you differentiate the outer function and multiply it by the derivative of the inner function. For instance, if a function involves \( e^{f(x)} \) rather than \( e^x \), the chain rule would be essential to correctly find the derivative by taking the exponential function's derivative and multiplying it by \( f'(x) \). Understanding the chain rule enhances flexibility in handling various compounded functions.