Problem 8 - Calculus for Biology and Medicine

Question

Differentiate the functions with respect to the independent variable. \(f(x)=e^{-3\left(x^{3}-1\right)^{4}}\)

Step-by-Step Solution

Answer

\(-36x^2 (x^3 - 1)^3 e^{-3(x^3 - 1)^4}\)

Step 1: Identify the Variable and Function Type

The task is to differentiate the function \( f(x) = e^{-3(x^3 - 1)^4} \) with respect to the independent variable \( x \). The function involves an exponential function with a composite argument, suggesting the use of chain rule.

Step 2: Apply the Chain Rule

The chain rule states that if \( y = u(v(x)) \), then \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx} \). Here, let \( u = e^v \) with \( v = -3(x^3 - 1)^4 \). The derivative of \( u \) with respect to \( v \) is \( e^v \), and the derivative of \( v \) will be computed next.

Step 3: Differentiate the Inner Function

To find \( \frac{dv}{dx} \), differentiate \( v = -3(x^3 - 1)^4 \) with respect to \( x \). First, apply the chain rule to differentiate \(-3(x^3-1)^4\):\(-3\cdot 4(x^3 - 1)^3\cdot\frac{d}{dx}(x^3 - 1) \).

Step 4: Differentiate the Innermost Function

The derivative of \( (x^3 - 1) \) with respect to \( x \) is \( 3x^2 \). Now substitute this back into the expression from Step 3. So, \( \frac{dv}{dx} = -12(x^3 - 1)^3(3x^2) \).

Step 5: Combine Results Using the Chain Rule

Using the results from the previous steps, find the derivative \( \frac{df}{dx} = e^{-3(x^3 - 1)^4} \cdot (-12)(x^3 - 1)^3 (3x^2) \). Simplify the expression to obtain \( \frac{df}{dx} = -36x^2 (x^3 - 1)^3 e^{-3(x^3 - 1)^4} \).

Key Concepts

Chain Rule

The chain rule is an essential tool in calculus for handling derivatives of composite functions. It allows us to break down complex functions into simpler components. Imagine you have two functions nested together, such as \( f(g(x)) \). The chain rule helps to find the derivative of such a function by multiplying the derivative of the outer function by the derivative of the inner function.
In mathematical terms, if \( y = f(u) \) where \( u = g(x) \), then the derivative \( \frac{dy}{dx} \) is \( \frac{dy}{du} \cdot \frac{du}{dx} \).
This concept is particularly powerful when dealing with exponential or trigonometric functions that are composed with polynomials or other functions.

• First, identify the outer and inner functions.
• Next, differentiate each function separately.
• Finally, multiply these derivatives to get the result.

Exponential Functions

Exponential functions are a pivotal part of calculus and appear frequently in real-world applications. These functions have the form \( f(x) = a^x \), where \( a \) is a constant, and \( x \) is the variable.
In the context of differentiation, one of the most fascinating elements of exponential functions is their unique property: the derivative of \( e^x \) is \( e^x \).
This property makes them quite simple to handle when differentiating, even if they appear in a more complex composition, such as in the function \( e^{-3(x^3 - 1)^4} \).
When differentiating exponential functions, particularly with compositions, chain rule often comes into play:

• Identify the exponent which generally is a function of \( x \).
• Differentiating with respect to \( x \) entails deriving the exponent first, then multiply by the derivative of the exponential function with respect to the exponent.

Composite Functions

Composite functions combine two or more functions, with one function serving as the input to another. In differentiation, they're handled by leveraging the chain rule. Consider \( f \) and \( g \) to be two functions; the composite function \( f(g(x)) \) requires a systematic approach to differentiate.
To differentiate a composite function:

• First, identify the outer function and the inner function.
• After identifying the functions, take the derivative of the outer function with respect to the inner function.
• Next, differentiate the inner function with respect to its variable.
• Multiply these derivatives according to the chain rule.

Applying this process to our exercise, \( f(x) = e^{-3(x^3 - 1)^4} \), the outer function is the exponential \( e^u \), and the inner function is \( u = -3(x^3 - 1)^4 \).
Differentiating these correctly gives us the complete derivative.

Derivative Calculation

Derivative calculation is the central process in calculus that allows us to determine rates of change, slopes, and behaviors of functions. Calculating derivatives involves rules and formulas, such as product rules, quotient rules, but one of the most versatile is the chain rule.
Evaluating the derivative of a function like \( f(x)=e^{-3(x^3-1)^4} \) requires a systematic approach to handle complex arguments within functions.

• First, decompose the function into simpler parts or layers.
• Calculate the derivative of each layer using known rules, such as how to differentiate powers \((ax^n)\), constants, and exponentials.

Finally, combine the derivatives constructively, respecting the relationships dictated by the function composition. The step-by-step handling ensures that whether encountering polynomial, exponential, or logarithmic layers, every contribution to the rate of change is accounted for and captured accurately. Remember to simplify results for cleaner, clearer answers.