Problem 54 - Calculus Single Variable

Question

A function \(f\) is given. Use logarithmic differentiation to calculate \(f^{\prime}(x)\). $$ f(x)=x^{\sin (x)} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = x^{\sin(x)} (\cos(x) \ln(x) + \frac{\sin(x)}{x}) \).

Step 1: Take the Natural Logarithm of Both Sides

Begin by taking the natural logarithm of both sides of the equation to simplify the differentiation process. This yields:\[ \ln(f(x)) = \ln(x^{\sin(x)}) \]Using the logarithmic identity \(\ln(a^b) = b \cdot \ln(a)\), we can rewrite the right side as:\[ \ln(f(x)) = \sin(x) \cdot \ln(x) \]

Step 2: Differentiate Implicitly

Differentiate both sides of the equation with respect to \(x\). On the left side, apply the derivative of \(\ln(f(x))\):\[ \frac{d}{dx}[\ln(f(x))] = \frac{f'(x)}{f(x)} \]On the right side, apply the product rule to \(\sin(x) \cdot \ln(x)\):\[ \frac{d}{dx}[\sin(x) \cdot \ln(x)] = \cos(x) \cdot \ln(x) + \sin(x) \cdot \frac{1}{x} \]

Step 3: Solve for \(f'(x)\)

Combine the derivatives from Step 2 and solve for \(f'(x)\). This will involve multiplying through by \(f(x)\) to clear the fraction:\[ \frac{f'(x)}{f(x)} = \cos(x) \cdot \ln(x) + \frac{\sin(x)}{x} \]\[ f'(x) = f(x) \left( \cos(x) \cdot \ln(x) + \frac{\sin(x)}{x} \right) \]

Step 4: Substitute Back for \(f(x)\)

Recall that \(f(x) = x^{\sin(x)}\). Substitute this back into the expression for \(f'(x)\):\[ f'(x) = x^{\sin(x)} \left( \cos(x) \cdot \ln(x) + \frac{\sin(x)}{x} \right) \]

Key Concepts

Derivative Calculation

Derivative calculation is a fundamental concept in calculus that involves finding the instantaneous rate of change of a function as the input variable changes. In simpler terms, it helps us understand how a function's output changes when the input is adjusted slightly. When tackling derivative calculations, there are a few important steps to keep in mind:

• Identify the function whose derivative you need to find.
• Determine the rules or techniques that apply, such as the power rule, chain rule, product rule, or quotient rule.
• Apply these rules systematically to find the derivative, often simplifying the expression where possible.

Derivative calculation is crucial for analyzing and predicting behaviors in various mathematical models, like growth rates or velocity. It is used extensively across different fields like physics and economics.

Implicit Differentiation

Implicit differentiation is a method used to find the derivative of a function when it is not explicitly solved for one variable in terms of another. This technique is particularly useful when dealing with equations where the variables cannot be easily separated, such as functions stated implicitly rather than in the standard explicit form y = f(x). To perform implicit differentiation, follow these steps:

• Differentiate both sides of the equation with respect to the independent variable. Don't forget to apply the chain rule to terms involving the dependent variable.
• Solve for the derivative you need. This might involve algebraic manipulation to isolate the derivative on one side of the equation.

Implicit differentiation is often used in cases involving trigonometric functions, exponential functions, and other complex expressions involving multiple variables.

Product Rule

The product rule is a handy differentiation technique used when dealing with functions that are the product of two or more separate functions. It allows you to find the derivative of the product of these functions by taking into account how each part changes.When applying the product rule, the derivative of a product of functions u(x) and v(x) is given by:\[ (u \, v)' = u' \, v + u \, v' \]Here's how to apply the product rule:

• Differentially each function individually.
• Multiply the derivative of the first function by the second function.
• Multiply the derivative of the second function by the first function.
• Add these results together to get the final derivative of the product.

This rule is essential when differentiating products where other straightforward rules like the power rule are not directly applicable.

Natural Logarithm

The natural logarithm is a logarithm with base e, where e is an irrational constant approximately equal to 2.71828. It is an essential tool in calculus, particularly when simplifying expressions for differentiation or integration. The natural logarithm has several key properties:

• ln(1) = 0: The natural logarithm of 1 is always 0, since any non-zero number raised to zero will equal 1.
• ln(e) = 1: Since e raised to the power of 1 is e.
• ln(a^b) = b ln(a): This property makes it easy to differentiate expressions involving powers.

Using natural logarithms simplifies the process of taking derivatives of complex functions, particularly those involving exponents or products. This simplification is central to the technique known as logarithmic differentiation, which was applied effectively in solving the given exercise.