Problem 22 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The derivative of \( f(x) = \cos(e^x) \) is \( f'(x) = -e^x \sin(e^x) \).

Step 1: Identify the Outer and Inner Functions

The function given is a composition of two functions: the outer function is the cosine function, \(g(u) = \cos(u)\), and the inner function is the exponential function, \(u(x) = e^x\).

Step 2: Differentiate the Outer Function

Differentiate the outer function \(g(u) = \cos(u)\) with respect to \(u\). The derivative is \(g'(u) = -\sin(u)\).

Step 3: Differentiate the Inner Function

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

Step 4: Apply the Chain Rule

Use the chain rule to differentiate the composite function \(f(x) = \cos(e^x)\). According to the chain rule, \(f'(x) = g'(u(x)) \cdot u'(x)\). Substitute the derivatives found in previous steps to get: \[ f'(x) = -\sin(e^x) \cdot e^x \].

Step 5: Simplify the Result

Simplify the expression for the derivative: \[ f'(x) = -e^x \sin(e^x) \].

Key Concepts

Understanding Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative measures how a function changes as its input changes.
Understanding derivatives is crucial because they tell us the rate at which a function is changing at any given point.
Key points to remember about differentiation:

• A derivative of a function gives us the slope of the tangent to the function's graph at any point.
• It is denoted by \( f'(x) \) for a function \( f(x) \), which reads as "the derivative of \( f \) with respect to \( x \)."
• Basic rules of differentiation include the power rule, product rule, quotient rule, and chain rule, among others.
• Differentiation helps understand phenomena in physics, engineering, economics, and others, wherever change needs to be analyzed.

Exploring Composite Functions

Composite functions are functions made by combining two or more functions, which means the output of one function becomes the input of another. In our example, we have a composite function \( f(x) = \cos(e^x) \), where an exponential function is inside a cosine function.
What to know about composite functions:

• The notation \( f(g(x)) \) indicates that \( g(x) \) is applied first, followed by \( f(x) \).
• To differentiate a composite function, you need to apply the chain rule, which is essential when dealing with compositions.
• In our example, \( \cos(e^x) \) is composite because \( e^x \) (an inner function) is contained within \( \cos() \) (outer function).

Differring sequential functions can be tricky, but understanding the role each plays clarifies the process. Differentiating composite functions often requires multiple steps, breaking down outer and inner functions separately, then combining them using the chain rule.

Derivative of Exponential Function

Exponential functions have the form \( e^x \), where \( e \) is the irrational number approximately equal to 2.71828. Exponential growth and decay describe change, making their understanding crucial in real-world applications.
Here's what is important about the derivative of an exponential function:

• The derivative of \( e^x \) with respect to \( x \) is remarkably simple: it stays the same, \( e^x \).
• This property makes \( e^x \) unique among functions because its rate of change is proportional to its value, leading to consistent exponential growth or decay.
• In solving the example problem, we observed how \( e^x \) helped construct our composite function and its derivative was seamlessly incorporated into the chain rule.

Exponential functions capture many processes where growth or decay depends on present quantity like population growth and radioactive decay, thus their derivatives provide profound insights into dynamic systems.