Question

Describing a Relationship In Exercises 99 and 100 , the relationship between \(f\) and \(g\) is given. Explain the relationship between \(f^{\prime}\) and \(g^{\prime} .\) $$ g(x)=f\left(x^{2}\right) $$

Short answer

From the given relationship \(g(x) = f(x^2)\), the relationship between their derivatives is \(g'(x) = 2x \cdot f'(x^2)\).

Step-by-step solution

Understand the Chain Rule

The chain rule is a basic derivative rule that states essentially that the derivative of the composition of two functions is the derivative of the outer function evaluated at the inner function, times the derivative of the inner function. In mathematical notations, given a function \(h(x)=f(g(x))\), its derivative \(h'(x)\) is given by \(h'(x)=f'(g(x)) \cdot g'(x)\). In this case, \(g(x)=f(x^2)\) can be seen as a function composition, where \(f\) is the outer function and \(x^2\) is the inner function.

Apply the Chain Rule

Apply the chain rule to \(g(x)\) to compute \(g'(x)\). By treating \(f\) as the outer function and \(x^2\) as the inner function \(u\), we have \(g'(x) = f'(u) \cdot u'(x)\), where \(u'(x)\) is the derivative of \(x^2\). The derivative of \(x^2\) is \(2x\), so \(g'(x) = f'(x^2) \cdot 2x\).

Final Relationship

From Step 2, the relationship between the derivatives has been found. The derivative of \(g(x)\) ,\(g'(x)\), is equal to the derivative of \(f(x)\) at \(x^2\) times \(2x\): \(g'(x) = f'(x^2) \cdot 2x\).

Key concepts

Derivatives of Composite Functions

In calculus, the concept of derivatives of composite functions is a cornerstone for understanding more advanced topics. It is akin to peeling an onion layer by layer, where each layer represents a function, and we want to understand the rate of change at each level.

Imagine we have two functions, one nested within the other. The outer function is like a machine that takes an input, processes it, and provides an output, while the inner function is another machine that first modifies the initial input. The derivative of this composite function tells us how the rate of change at the outer level is affected by the rate of change at the inner level.

An Example with the Chain Rule

Consider the function given in the textbook example: \( g(x) = f(x^2) \). To find the derivative \( g'(x) \), we must account for both the outer function \( f \) and the inner function \( x^2 \). Applying the Chain Rule, we first find the derivative of the outer function with respect to its input, \( f'(u) \) where \( u \) represents the inner function. Then, we multiply it by the derivative of the inner function, \( u'(x) \) which in this case is \( 2x \). The resulting expression \( g'(x) = f'(x^2) \cdot 2x \) encapsulates the intertwined rates of change.

Chain Rule Applications

Real-World Scenarios

The Chain Rule isn't just an abstract mathematical theorem; it has numerous real-world applications. Whenever we have a situation involving multiple layers of processes or changes, the Chain Rule can come into play.

For instance, in physics, we often encounter quantities that depend on each other in a chain-like manner, such as in the case of displacement, velocity, and acceleration. Similarly, in economics, the reaction of market demand to raw material costs often follows a composite function relationship.

In the given exercise, the Chain Rule helps us to connect the relationships between \( f' \) and \( g' \) succinctly. This understanding is not just crucial for solving textbook problems, but also for interpreting complex relationships in fields like engineering, economics, and the natural sciences.

Differentiation Techniques

Differentiation techniques encompass a variety of rules and methods used to calculate the derivative of functions. Apart from the Chain Rule, these techniques include the Product Rule, the Quotient Rule, and the Power Rule, among others. Each rule applies to different kinds of function constructions.

Understanding When to Use Each Rule

To master differentiation techniques, it is essential to recognize the structure of the function you are dealing with. For a product of functions, use the Product Rule; for a ratio, the Quotient Rule; and for functions raised to a power, the Power Rule is your friend.

The Chain Rule, in particular, is essential when dealing with nested functions or functions within functions, as in our textbook example. Developing an intuition for which technique applies gives you a powerful toolset for solving calculus problems. Moreover, these techniques aren't just academic exercises; they are the foundation for many algorithms and models in science and engineering.