Problem 21 - Calculus for Biology and Medicine

Question

Differentiate the functions with respect to the independent variable. \(f(x)=\sin \left(e^{x}\right)\)

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \cos(e^x) \cdot e^x \).

Step 1: Identify the Outer and Inner Functions

To differentiate the function \(f(x) = \sin(e^x)\), we first identify the outer and inner functions. The outer function is \( \sin(u) \), where \( u = e^x \). The inner function is \( u = e^x \).

Step 2: Differentiate the Outer Function

Differentiate the outer function \( \sin(u) \) with respect to \( u \). The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \).

Step 3: Differentiate the Inner Function

Differentiate the inner function \( u = e^x \) with respect to \( x \). The derivative of \( e^x \) with respect to \( x \) is \( e^x \).

Step 4: Apply the Chain Rule

Use the chain rule to find the derivative of \( f(x) = \sin(e^x) \). According to the chain rule, \( \frac{df}{dx} = \frac{d}{du} \sin(u) \cdot \frac{du}{dx} \). Substituting the derivatives we found: \( \frac{df}{dx} = \cos(e^x) \cdot e^x \).

Step 5: Write the Final Derivative

The derivative of the function \( f(x) = \sin(e^x) \) with respect to \( x \) is \( f'(x) = \cos(e^x) \cdot e^x \). This is the final result.

Key Concepts

Understanding the Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. A composite function is a function made up of two or more functions. In our case, we have a function nested within another, like \(f(x) = \sin(e^x)\). To differentiate such functions, the chain rule provides a handy formula.

• The chain rule states that if you have a function \(y = f(g(x))\), then the derivative \(\frac{dy}{dx}\) is given by \(\frac{dy}{dg} \times \frac{dg}{dx}\).
• Simply put, differentiate the outer function and then multiply it by the derivative of the inner function.

This method simplifies the process of tackling composite functions, ensuring accuracy in calculations.

Identifying the Outer Function

The outer function is the function that is applied last in a composite function, visualized as the "big picture" or the "outside layer" of our equation. In the function \(f(x) = \sin(e^x)\), the outer function is \(\sin(u)\), where \(u\) is substituted for \(e^x\).

• To differentiate the outer function, think about how it behaves independently.
• For \(\sin(u)\), its derivative is \(\cos(u)\).

This outer perspective is key to simplifying the differentiation process, making it more coherent as you deal with layers of complexity.

Discovering the Inner Function

Before applying the chain rule, we must pinpoint the inner function. This is the function applied first, or the "inside layer" of our composite function. In \(f(x) = \sin(e^x)\), the inner function is clearly \(u = e^x\).

• Think of the inner function as the core of the composite function.
• Its derivative is crucial since it changes how the outer function behaves when composed.
• For our given function, the derivative of \(u = e^x\) with respect to \(x\) is simply \(e^x\).

Having a firm grasp of the inner function allows for a straightforward execution of the chain rule.

Mastering Derivative Calculation

Differentiation involves computing the rate at which one quantity changes in relation to another. In our problem, we calculate how \(\sin(e^x)\) changes as \(x\) varies. By deploying the chain rule, we shift smoothly from individual derivatives to the overall derivative.

• Start with the derivatives obtained from the outer and inner functions.
• Combine these differentiated components to find \(f'(x)\).
• The formula, derived using the chain rule, becomes \(f'(x) = \cos(e^x) \cdot e^x\).
• This result illustrates the joined impact of both the outer and inner functions on \(f(x)\).

The comprehensive understanding of derivative calculation provides the final link that bridges differentiation concepts, empowering you to tackle similar problems confidently.