Problem 20 - Calculus for Biology and Medicine

Question

Differentiate the functions with respect to the independent variable. \(f(x)=\exp \left[\cos \left(1-2 x^{3}\right)\right]\)

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 6x^2 e^{\cos(1 - 2x^3)} \cdot \sin(1 - 2x^3) \).

Step 1: Identify the Structure

The given function is a composition of several functions. Recognize it as an exponential function where the exponent is \( [v(x) = \cos(1 - 2x^3)]\), thus the function can be rewritten as \( f(x) = e^{v(x)}\).

Step 2: Apply the Chain Rule

To differentiate \( f(x) = e^{v(x)}\), use the chain rule. This gives the derivative as:\[ f'(x) = e^{v(x)} \cdot v'(x)\]where \( v'(x)\) is the derivative of the inner function \( v(x)\).

Step 3: Differentiate the Inner Function

Since \( v(x) = \cos(1 - 2x^3)\), apply the chain rule again to find \( v'(x)\). Differentiating \( \cos(1 - 2x^3)\) with respect to \( x\) yields:\[ v'(x) = -\sin(1 - 2x^3) \cdot (1 - 2x^3)'\]Next, differentiate \( 1 - 2x^3 \) to get \(-6x^2\).

Step 4: Combine Results

Substitute \(v'(x)\) back into the expression for \( f'(x)\):\[ f'(x) = e^{\cos(1 - 2x^3)} \cdot [-\sin(1 - 2x^3) \cdot (-6x^2)]\]Simplify the expression to get the final derivative:\[ f'(x) = 6x^2 \cdot e^{\cos(1 - 2x^3)} \cdot \sin(1 - 2x^3)\]

Step 5: Simplification Check

Verify that the expression is as simple as possible. The expression cannot be further simplified algebraically, so this is the final derivative.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It involves finding the derivative of a function, which tells us how the function changes as its input changes. This is crucial for understanding the behavior of functions at any given point. When differentiating, we essentially want to calculate the rate of change of the function's output with respect to changes in its input.

To properly differentiate a function, it is important to identify its components and apply the appropriate rules. In this case, the chain rule is a primary tool used for differentiating composite functions. The chain rule states that if we have a composite function, say \( f(g(x)) \), its derivative can be found by multiplying the derivative of \( f \) with respect to \( g(x) \) by the derivative of \( g(x) \) with respect to \( x \). This is expressed mathematically as:

• \( (f(g(x)))' = f'(g(x)) \, g'(x) \)

Using differentiation allows us to solve a wide array of real-world problems, such as finding the speed of a moving object or understanding how changing one variable affects another in scientific experiments.

Exponential Function

Exponential functions like \( e^x \) are functions where the variable is in the exponent. The base of the natural exponential function is \( e \), an irrational number approximately equal to 2.71828. Exponential functions have distinct properties compared to polynomial and linear functions. They grow rapidly as the input increases, which makes them very powerful for modeling processes like population growth and radioactive decay.

A unique feature of the exponential function is its derivative. For the natural exponential function \( e^x \), the derivative is itself:

• \( \frac{d}{dx} e^x = e^x \)

When dealing with more complex exponential functions like \( e^{v(x)} \), where \( v(x) \) is another function, we must use the chain rule. This requires us to differentiate \( v(x) \) first, and then multiply that derivative by the exponential itself. This characteristic makes the exponential function a common choice in calculus problems involving growth and decay, as well as financial applications like compound interest.

Composite Function

Composite functions combine two or more functions into one using composition. Given functions \( f(x) \) and \( g(x) \), their composition \( f(g(x)) \) means that \( g(x) \) is substituted into \( f \). This results in a function that inherits properties from both \( f \) and \( g \). Composite functions allow us to express complex relationships succinctly.

In the given exercise, our function is composed of several layers: an exponential function and a trigonometric function embedded within, specifically represented as \( e^{\cos(1 - 2x^3)} \). To differentiate such a function, breaking it down into its components is necessary. Applying the chain rule not once, but multiple times, is usually needed.

This begins with differentiating the outermost function while keeping the inner function intact, followed by progressively unraveling each inner function. A vital step is recognizing each layer correctly, as errors here can lead to incorrect derivatives. Understanding this multi-layered process is vital for anyone studying calculus, as it extends to understanding complex systems in engineering, physics, and beyond.