Problem 23 - Calculus for Biology and Medicine

Question

In Problems 1-28, 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: Identify the function and rewriting

The function given is \( f(x) = \sqrt[7]{x^2 - 2x + 1} \). Before finding the derivative, rewrite it in a power form: \( f(x) = (x^2 - 2x + 1)^{1/7} \). This makes it easier to apply the chain rule.

Step 2: Use the chain rule for differentiation

The chain rule states that if you have a composite function \( g(h(x)) \), the derivative \( f'(x) \) is \( g'(h(x)) \cdot h'(x) \). Here, let \( u = x^2 - 2x + 1 \), making \( f(x) = u^{1/7} \). The derivative is \( f'(x) = \frac{1}{7}u^{-6/7} \cdot \frac{du}{dx} \).

Step 3: Differentiate the inner function

Now, find the derivative of the inner function \( u = x^2 - 2x + 1 \). The derivative of \( u \) is \( \frac{du}{dx} = 2x - 2 \).

Step 4: Substitute back and simplify

Substitute \( u \) and its derivative back into the expression for \( f'(x) \). You get: \( f'(x) = \frac{1}{7}(x^2 - 2x + 1)^{-6/7} \cdot (2x - 2) \). Simplify further: \( f'(x) = \frac{2x - 2}{7(x^2 - 2x + 1)^{6/7}} \).

Key Concepts

Understanding the Chain Rule

One of the key techniques in calculus differentiation is the chain rule. It's a method used to compute the derivative of a composite function. Consider a composite function in the form of \( g(h(x)) \). Here, the function \( g \) is applied to the output of the function \( h \). The chain rule lets us find the derivative of \( g(h(x)) \) with ease by following a particular formula. The rule states:

• If \( y = g(h(x)) \), then the derivative \( \frac{dy}{dx} = g'(h(x)) \cdot h'(x) \).

To break it down, you'll first need to:

• Differentiate the outer function \( g \) with respect to its input, \( h(x) \), giving you \( g'(h(x)) \).
• Next, differentiate the inner function \( h(x) \) with respect to \( x \), giving you \( h'(x) \).
• Finally, multiply these two derivatives together to obtain the desired derivative of the composite function.

Practicing with the chain rule repeatedly helps in managing complex expressions, allowing the unraveling of layered functions step by step.

Derivative of Composite Functions

Now let's delve into the concept of composite functions themselves. A composite function is formed by combining two functions such that the output of one function becomes the input of another. For example, in the composite function \( f(x) = \sqrt[7]{x^2 - 2x + 1} \), the main function is the 7th root, whereas the inner function is the quadratic expression \( x^2 - 2x + 1 \).

• First step in differentiation is to rewrite this composite function in a format that can be more easily differentiated. It's typically converted into an exponent form, like \( f(x) = (x^2 - 2x + 1)^{1/7} \). This setup allows for straightforward application of differentiation rules.
• Using the chain rule as described above, we differentiate first the outer function with respect to the inner function, and then multiply by the derivative of the inner function itself.
• Thus, the differentiation process breaks down into simpler steps, making calculation manageable and less prone to error.

Clear understanding of how composite functions interact and being able to identify the inner and outer components skilfully is crucial for mastering the chain rule.

Exploring the Power Rule in Calculus

The power rule is one of the foundational tools in differentiation, facilitating the process of deriving powers of a variable. If you have a power function \( x^n \), the derivative of this function is straightforwardly found using the power rule:

• For \( y = x^n \), the derivative \( \frac{dy}{dx} = nx^{n-1} \).

In the problem we solved, the function \( f(x) = (x^2 - 2x + 1)^{1/7} \) applies the power rule in a slightly more complex scenario:

• Here, \( n = \frac{1}{7} \), so when we differentiate, we bring the exponent down and subtract one from the exponent to get \( \frac{1}{7}(x^2 - 2x + 1)^{-6/7} \).
• We then multiply this result by the derivative of the inner function as per the chain rule.
• This usage of the power rule allows us to cleanly and accurately solve derivative problems that might initially appear complex.

Having a strong grip on the power rule is indispensable for those venturing into the deeper realms of calculus, providing a straightforward stepping stone to more advanced topics.