Problem 72 - Calculus Single Variable

Question

If \(f\) is a differentiable function, find a formula for \(\left(f^{2}\right)^{\prime}(x)\). Next find a formula for \(\left(f^{3}\right)^{\prime}(x) .\) What do you expect the formula for \(\left(f^{n}\right)^{\prime}(x)\) is? (Predict by analogy, but do not prove this last formula. These matters will be taken up later in the chapter.)

Step-by-Step Solution

Answer

For \((f^n)'(x)\), the formula is \(n f^{n-1}(x) f'(x)\).

Step 1: Recognize the Function Composition

The function given is a composition of functions, specifically a power of the function \( f(x) \). When we have \( f^n(x) \), it can be expressed as \( (f(x))^n \). This implies that to differentiate it with respect to \( x \), we will need to use the chain rule in conjunction with the power rule.

Step 2: Differentiate \( (f^2)^{\prime}(x) \)

We express \( (f^2)(x) \) as \( (f(x))^2 \). To find \( (f^2)^{\prime}(x) \), apply the chain rule: for \( u(x) = f(x) \), \( (u(x))^2 \) derives as \( 2u(x) \, u'(x) \). Substitute back to get:\[(f^2)'(x) = 2f(x)f'(x)\]

Step 3: Differentiate \( (f^3)^{\prime}(x) \)

Similarly, \( (f^3)(x) = (f(x))^3 \). Apply the derivative again using the chain rule: \( 3(f(x))^2 \, f'(x) \). Hence, we have:\[(f^3)'(x) = 3f^2(x)f'(x)\]

Step 4: Make an Analogy for \( (f^n)^{\prime}(x) \)

Observing the previous results, the pattern emerges. For \( f^2(x) \), the derivative involves \( 2f(x) \, f'(x) \); for \( f^3(x) \), it is \( 3f^2(x) \, f'(x) \). By analogy, this pattern suggests the general rule:\[(f^n)'(x) = n f^{n-1}(x) f'(x)\]

Key Concepts

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate compositions of functions. It allows us to compute the derivative of a function based on the derivatives of its component functions. When a function is composed of another function, such as an outer and inner function, the chain rule states that you must first differentiate the outer function, keep the inner function unchanged, and then multiply by the derivative of the inner function.

Here’s a step-by-step breakdown:

• Identify the outer and inner functions: If you have a function like \((g(f(x)))\), the chain rule separates it into an outer function \(g\) and an inner function \(f\).
• Differentiate the outer function: Calculate the derivative of \(g\) with respect to \(f\), leaving \(f\) unchanged.
• Multiply by the derivative of the inner function: Now differentiate \(f\) with respect to \(x\), and multiply this by the previous step's result.

When applied to functions like \((f(x))^n\), the chain rule becomes especially useful, as it helps connect the dots between the power and its underlying function structure.

Power Rule

The power rule is a simple but vital technique in differentiation that applies when a function is expressed in the form of a variable raised to a power. The essence of the power rule is to reduce the exponent by one and multiply the entire function by the original power.

Here’s the rule:

• Identify the function format: If you have a function \(u(x)^n\), it matches the power rule’s criterion.
• Differentiate using the rule: Take the exponent \(n\), multiply the function by it, and then subtract 1 from the exponent. This gives you \(nu(x)^{n-1}\).
• Combine with the chain rule: Often, especially in cases where \(u(x)\) is not simply \(x\), the power rule needs to be paired with the chain rule.

This rule simplifies how we differentiate polynomials and other functions when combined with more complex rules like the chain rule.

Function Composition

Function composition is a key concept, particularly relevant when dealing with functions within functions. It is the process of applying one function to the results of another, denoted as \(g(f(x))\) where \(f\) is applied first and \(g\) applies to the output of \(f\).

Understanding function composition:

• Recognize the nested function layers: In \((f(x))^n\) composition, \(f(x)\) is wrapped in an exponential form.
• Focus on applying composition rules: Each function layer must be tackled methodically, typically starting with the innermost one and moving outward.
• Utilize rules like chain rule: This is essential in differentiating compositions since multiple functions' derivatives interplay.

The genius of function compositions lies in their versatility and ability to build complex mathematical models from simple functions, making them invaluable in calculus and beyond.