Problem 81 - Brief Applied Calculus

Question

Suppose that \(L(x)\) is a function such that \(L^{\prime}(x)=\frac{1}{x} .\) Use the Chain Rule to show that the derivative of the composite function \(L(g(x))\) is \(\frac{d}{d x} L(g(x))=\frac{g^{\prime}(x)}{g(x)} .\)

Step-by-Step Solution

Answer

The derivative of \( L(g(x)) \) is \( \frac{g'(x)}{g(x)} \).

Step 1: Recall the Chain Rule

The Chain Rule states that if you have a composite function, say \( f(g(x)) \), then its derivative is determined by \( \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \). This formula will guide us through finding the derivative of the composite function \( L(g(x)) \).

Step 2: Apply the Chain Rule

Applying the Chain Rule to the given function \( L(g(x)) \), we obtain:\[\frac{d}{dx} L(g(x)) = L'(g(x)) \cdot g'(x).\] Here, \( L'(g(x)) \) represents the derivative of \( L \) at \( g(x) \).

Step 3: Substitute the Derivative of L

We know from the problem statement that \( L'(x) = \frac{1}{x} \). Now substitute \( g(x) \) in place of \( x \):\[L'(g(x)) = \frac{1}{g(x)}.\]

Step 4: Complete the Derivative Calculation

Substitute \( L'(g(x)) = \frac{1}{g(x)} \) back into the chain rule expression:\[\frac{d}{dx} L(g(x)) = \frac{1}{g(x)} \cdot g'(x).\]Simplifying, we get:\[\frac{d}{dx} L(g(x)) = \frac{g'(x)}{g(x)}.\] This confirms the original statement we set out to prove.

Key Concepts

Composite Function

A composite function is a function that is made by combining two or more functions. It involves applying one function to the results of another. For example, if you have functions \( f(x) \) and \( g(x) \), the composite function \( L(g(x)) \) essentially means you first apply \( g(x) \) and then apply \( L(x) \).

This is quite useful in calculus since it allows us to express complex operations in terms of simpler, nested functions. When dealing with composite functions, one of the crucial steps in calculus is finding their derivatives, which often requires a particular technique called the Chain Rule.

Derivative

The derivative of a function is a concept in calculus that represents an instantaneous rate of change. To understand it simply, the derivative tells us how a function is changing at any given point. It's like looking at the slope of a curve at a specific point.

If you have a simple function \( y = x^2 \), the derivative would be \( y' = 2x \), meaning for any value of \( x \), the derivative tells us the rate of change or slope at that point on the graph.

• Understanding derivatives is key to solving many calculus problems.
• They are often used to find slopes, rates of change, and optimize formulas in various fields.

The concept of derivatives gets more interesting with composite functions, where the Chain Rule becomes necessary.

Calculus

Calculus is a branch of mathematics that studies how things change. It provides tools for analyzing changes and patterns in functions. Calculus is split into two main parts: differential calculus and integral calculus.

Differential calculus focuses on the concept of derivatives, which help us understand how a function is changing at any point. This part of calculus allows you to compute things such as slopes and octane factors.

• Integral calculus, on the other hand, deals with accumulations, like the area under curves.
• Both branches together form the foundation of calculus, offering various techniques to solve real-world problems involving change.

One essential technique in differential calculus is differentiating complex or composite functions, where the Chain Rule plays a major role.

Function Differentiation

Function differentiation is the process of finding the derivative of a function. Differentiation tells us how a function changes in response to changes in its input.

For a simple function like \( y = x^3 \), the process involves applying rules like the power rule to get its derivative \( y' = 3x^2 \). For a composite function, however, it becomes a bit more complex.

• Here, we use the Chain Rule, which allows us to break down the derivative process into simpler parts.
• To differentiate a composite function \( L(g(x)) \), first find \( L'(g(x)) \), then multiply it by the derivative \( g'(x) \).

Differentiation, especially with functions inside of other functions, adds layers of complexity that are all well handled with the right tools, like the Chain Rule.