Problem 255 - Calculus Volume 1

Question

If \(f=g \circ h,\) when reversing the chain rule, \(\frac{d}{d x}(g \circ h)(x)=g^{\prime}(h(x)) h^{\prime}(x), \quad\) should you take \(u=g(x)\) or \(u=h(x) ?\)

Step-by-Step Solution

Answer

Take \(u = h(x)\).

Step 1: Understanding the Function Composition

The exercise deals with a function composition where \(f = g \circ h\). This means \(f(x) = g(h(x))\). This is important because it defines how the functions are nested.

Step 2: Analyzing the Chain Rule

The chain rule for differentiation states that if you have a composite function \((g \circ h)(x)\), its derivative is given as \(\frac{d}{d x}(g \circ h)(x)=g^{\prime}(h(x)) h^{\prime}(x)\). This form arises from treating \(h(x)\) as an inner function and \(g(u)\) as the outer function where \(u = h(x)\).

Step 3: Determining the Variable \(u\)

In the context of the chain rule, \(u\) is typically the inner function when breaking down the differentiation of a composite function. Therefore, given \(f = g \circ h\), the inner function is \(h(x)\), meaning \(u = h(x)\).

Step 4: Conclusion on Variable Choice

Therefore, since \(h(x)\) is the inner function applied inside \(g\), the correct choice for \(u\) when applying the chain rule in reverse is \(u = h(x)\).

Key Concepts

Composite Functions

In mathematics, a composite function is a fancy way of linking two functions together. Think of it like nesting one function inside another. If we have two functions, say \(g(x)\) and \(h(x)\), composing them would mean creating a new function \(f(x) = g(h(x))\).

This means we're plugging the value of \(x\) into \(h(x)\) first, and then taking the result and plugging it into \(g(x)\). This is very similar to a two-step process:

• First: Calculate \(h(x)\).
• Second: Use the result from \(h(x)\) and find \(g(h(x))\).

Composite functions are often written as \(f = g \circ h\), showcasing how the outputs of one function become inputs for the other, creating a chain of processes.

Function Composition

Function composition is like following a recipe where the result of one step is needed in the next. When you compose two functions, \(g\) and \(h\), and say it's \(g \circ h\), you're basically saying you want to perform \(h\) first, then \(g\) on the result of \(h\).

Visualizing function composition can help: imagine \(h(x)\) as a machine that modifies \(x\) in a specific way, and then \(g(x)\) further modifies the output from \(h(x)\). The formula for composition, \(f(x) = g(h(x))\), ensures we follow the right order and end up with the correct result.

• Start with \(x\).
• Apply \(h\) to \(x\).
• Then, apply \(g\) to the result of \(h(x)\).

Understanding this order and process is crucial when dealing with more complex operations, such as differentiation.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding how a function changes as its input changes. For composite functions, the chain rule is a critical tool. It helps in differentiating functions where one function is plugged into another.

Using the chain rule means you need to identify two parts of your composition:

• The "outer function", \(g\), which comes after.
• The "inner function", \(h\), the one done initially.

The chain rule formula is: \(\frac{d}{dx}(g(h(x))) = g'(h(x)) \cdot h'(x)\). This notation of differentiation tells you to:

• Differentiate the outer function first, while keeping the inner function \(h(x)\) unchanged.
• Next, multiply that by the derivative of the inner function.

By recognizing \(h(x)\) as the inner part, we know \(u = h(x)\) in terms of the variable choice. This choice simplifies applying the chain rule correctly.