Question

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ f(x)=\sqrt{x^{2}-4 x+2} $$

Short answer

\[f'(x) = \frac{x - 2}{\sqrt{x^{2} - 4x + 2}}\]

Step-by-step solution

Convert to a Simpler Form

Rewrite the function in a form that applies to the power rule in calculus. This becomes: \(f(x) = (x^{2} - 4x + 2)^{1/2}\)

Apply the Chain Rule

Differentiate using the chain rule: \[f'(x) = \frac{1}{2} (x^{2} - 4x + 2)^{-1/2} (2x - 4)\]

Simplify the Result

Simplify the derivative to its simplest form: \[f'(x) = \frac{x - 2}{\sqrt{x^{2} - 4x + 2}}\]

Key concepts

Understanding the Chain Rule

The chain rule is a powerful tool in calculus that allows us to differentiate composite functions, which are functions made up of other functions. It simplifies the process of finding derivatives when you have a combination of functions nested within each other.
To visualize the chain rule, let's imagine peeling an onion. Each layer represents a different function, and you need to differentiate each layer separately while following through to the next. When we differentiate the outer function, we hold the inside expression the same, then multiply the result by the derivative of the inner function.
In our given function, \(f(x) = \sqrt{x^2 - 4x + 2}\), we can treat \(g(x) = x^2 - 4x + 2\) as the inner function and \(u = \sqrt{u}\) (or \(u^{1/2}\)) as the outer function. By applying the chain rule, we first differentiate \(u^{1/2}\) with respect to \(u\) to get \(\frac{1}{2}u^{-1/2}\). Then we multiply it by the derivative of the inner function \(g(x)\), which is \(2x - 4\).
Use:

• Identify the outer and inner functions.
• Differentiate the outer function, keeping the inner function unchanged.
• Multiply by the derivative of the inner function.

This results in the derivative \(f'(x) = \frac{1}{2} (x^2 - 4x + 2)^{-1/2} (2x - 4)\).

Applying the Power Rule

The power rule is one of the most fundamental rules for differentiation, used when differentiating functions with exponents. It states: if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\). This rule simplifies the process of finding derivatives for polynomial terms.
To transform our function \(f(x) = \sqrt{x^2 - 4x + 2}\) into a form suitable for the power rule, we rewrite it with a fractional exponent: \(f(x) = (x^2 - 4x + 2)^{1/2}\). This transformation is crucial because it allows us to directly apply the power rule to differentiate the outer component of the chain.
Considerations:

• Ensure the function is expressed with an exponent before using the power rule.
• Differentiate by multiplying the power by the coefficient and reducing the power by one.

After applying the power rule as part of the chain rule process, the outer derivative turns into \(\frac{1}{2}(x^2 - 4x + 2)^{-1/2}\). It is through this newfound expression that easier simplification follows.

The Art of Simplifying Expressions

Simplifying expressions is a key step in calculus that ensures the derivative is a cleaner and more understandable form. After performing derivative calculations, the results may appear complex or cumbersome. Simplification helps remove extraneous factors and present the expression clearly.
In the problem, applying the chain rule gives the expression \(f'(x) = \frac{1}{2} (x^2 - 4x + 2)^{-1/2} (2x - 4)\). Here, simplification involves combining like factors and restructuring the expression to a tidier form:
- Distribute the numerator to remove the fraction.- Recognize symmetrical features that simplify the function further.- Arrange terms for clarity.
Steps:

• Perform any arithmetic to rearrange and combine terms.
• Cancel common factors if possible.
• Rewrite the expression to be more approachable.

Ultimately, simplifying the derivative expression yields \(f'(x) = \frac{x - 2}{\sqrt{x^2 - 4x + 2}}\). This final form is easier to use and interpret for further analysis or computation.