Problem 23 - Calculus for Biology and Medicine

Question

Differentiate the functions with respect to the independent variable. \(f(x)=\sqrt[7]{x^{2}-2 x+1}\)

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{2x - 2}{7(x^2 - 2x + 1)^{6/7}} \).

Step 1: Rewrite the Function

The given function is written under a radical form, which may not be the easiest form for differentiation. Rewrite the function as: \ \( f(x) = (x^2 - 2x + 1)^{1/7} \). This allows us to apply the power rule in differentiation.

Step 2: Apply the Chain Rule

To differentiate \(f(x) = (x^2 - 2x + 1)^{1/7}\), use the chain rule. The chain rule states \( \frac{du^n}{dx} = n \, u^{n-1} \, \frac{du}{dx} \) where \(u = x^2 - 2x + 1\). Here, \(n = \frac{1}{7}\), and now we need to find \( \frac{du}{dx} \).

Step 3: Differentiate the Inside Function

Find \( \frac{du}{dx} \) where \( u(x) = x^2 - 2x + 1 \). Differentiate using the sum and power rules: \ \( \frac{du}{dx} = \frac{d}{dx}(x^2) - \frac{d}{dx}(2x) + \frac{d}{dx}(1) = 2x - 2 \).

Step 4: Differentiate the Outer Function

Apply the chain rule: \( f'(x) = \frac{1}{7} (x^2 - 2x + 1)^{-6/7} \cdot (2x - 2) \). Simplify by substituting \( \frac{du}{dx} \) into the chain rule.

Step 5: Simplify the Derivative

Simplify the expression for \( f'(x) \): \ \( f'(x) = \frac{1}{7} (x^2 - 2x + 1)^{-6/7} (2x - 2) \). \That's the derivative of the original function with respect to \(x\).

Key Concepts

Differentiation

Differentiation is one of the core concepts in calculus. It involves finding the derivative of a function, which helps us understand how a function's output changes with respect to changes in its input. Essentially, the derivative provides the rate of change or the slope of a function at any given point.
When differentiating a function, we apply various rules and techniques to handle different types of equations. Understanding differentiation is crucial for tackling problems related to rates of motion, optimization, and the behavior of physical systems. Using the chain rule method is often necessary when a function is composed of other functions, as it helps us differentiate more complex expressions effectively. This concept is well exemplified in the provided exercise where the function, originally presented in radical form, is re-expressed for easier manipulation.

Power Rule

The power rule is a fundamental tool in differentiation used to find the derivative of polynomial expressions. It states that if you have a function in the form of a power, such as \(x^n\), its derivative is \(nx^{n-1}\).
This makes handling polynomial differentiations straightforward. For instance, in the exercise, after rewriting the function under a power form, we were able to easily apply the power rule along with the chain rule. This allows us to efficiently differentiate the outer part of the function \((x^2 - 2x + 1)^{1/7}\). Understanding the power rule is crucial for dealing with more complex calculus problems as it lays the groundwork for applying more advanced differentiation rules.

Calculus

Calculus is a branch of mathematics that explores continuous change, and it's divided mainly into two parts: differentiation and integration. In this exercise, we focused on the differentiation part, which deals with finding the rate at which things change.
Calculus allows us to model and solve real-world problems involving dynamic systems, such as physics or engineering challenges. By using calculus and its rules like the chain and power rules, we can differentiate complex functions and obtain derivatives.
Derivatives are essential in predicting future behaviors and understanding the dynamics of various systems in nature, economics, and beyond. This is why mastering calculus fundamentals is so important—it gives you powerful tools to solve intricate problems.