Problem 82 - Brief Applied Calculus

Question

Suppose that \(E(x)\) is a function such that \(E^{\prime}(x)=E(x)\). Use the Chain Rule to show that the derivative of the composite function \(E(g(x))\) is \(\frac{d}{d x} E(g(x))=E(g(x)) \cdot g^{\prime}(x) .\)

Step-by-Step Solution

Answer

The derivative of \(E(g(x))\) is \(E(g(x)) \cdot g'(x)\).

Step 1: Understand the Given Function

We know that the function \(E(x)\) has the property that its derivative is equal to the function itself, i.e., \(E^{\prime}(x) = E(x)\). This implies \(E(x)\) is an exponential function.

Step 2: Apply the Chain Rule

The Chain Rule states that if you have a composite function \(E(g(x))\), the derivative with respect to \(x\) is given by \(\frac{d}{dx}[E(g(x))] = E^{\prime}(g(x)) \cdot g^{\prime}(x)\).

Step 3: Substitute the Known Derivative

Since \(E^{\prime}(x) = E(x)\), we substitute \(E^{\prime}(g(x))\) with \(E(g(x))\). Thus, the derivative becomes \(E(g(x)) \cdot g^{\prime}(x)\).

Step 4: Conclusion of Steps

We have found that using the Chain Rule and the given property of \(E(x)\), the derivative of \(E(g(x))\) is \(E(g(x)) \cdot g^{\prime}(x)\). This confirms the relationship we needed to show.

Key Concepts

Chain Rule

The Chain Rule is a fundamental theorem in calculus used to compute the derivative of a composite function. In simpler terms, it's a rule that allows us to differentiate functions that are "nested" inside one another. To understand this, think of a composite function as one function inside another, like a Russian nesting doll.

Imagine you have a function described as \(f(g(x))\). The Chain Rule tells us how to differentiate this composite function with respect to \(x\). The formula is:

• First, differentiate the outer function \(f\) with respect to its inner function (let's call it \(u\)), giving \(f'(u)\).
• Then, multiply by the derivative of the inner function \(g(x)\) with respect to \(x\), giving \(g'(x)\).

This results in the Chain Rule formula for \(f(g(x))\) as follows: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\).
In our original exercise, \(E(g(x))\) is a composite function, with \(E\) as the outer function and \(g(x)\) as the inner function. Applying the Chain Rule efficiently breaks down the differentiation process.

Composite Functions

Composite functions are functions constructed by combining two or more functions. In notation, if you have two functions \(f(x)\) and \(g(x)\), their composition is denoted as \(f(g(x))\). This means you first apply \(g\) to \(x\), and then apply \(f\) to the result from \(g(x)\).

This process can be visualized as processing an input through multiple stages. For example:

• First, you input \(x\) into \(g\).
• Second, the output of \(g(x)\) becomes the input for \(f\).

With the composition \(E(g(x))\), \(g(x)\) is acted upon first, and the result is then processed by \(E\).
This layered approach is fundamental in advanced calculus, as it enables the building of complex equations from simpler ones. Understanding composite functions is crucial as they naturally lead us to utilize the Chain Rule for differentiation.

Exponential Functions

Exponential functions are a special class of functions where a constant base is raised to a variable exponent, typically described in the form \(b^{x}\). A common exponential function seen in calculus is the natural exponential function, represented as \(e^{x}\), where \(e\) is approximately 2.71828.

In the context of differentiation, exponential functions have a unique quality: the derivative of \(e^{x}\) is itself, \(e^{x}\). Therefore, if we have an exponential function \(E(x)\) such that \(E'(x) = E(x)\), it clearly aligns with the properties of the natural exponential.

Our initial problem states that \(E'(x) = E(x)\), indicating that \(E(x)\) is an exponential function—it has the same form as \(e^{x}\).

• When \(E(x)\) is used within a composite function as in \(E(g(x))\), this self-deriving property becomes powerful.
• It simplifies the calculation by maintaining the same form, even after differentiation, as long as the Chain Rule is properly applied.

Understanding exponential functions and their differentiation is key to solving complex derivative problems efficiently.