Question

Express the indicated derivative in terms of the function \(F(x) .\) Assume that \(F\) is differentiable. $$ D_{x}\left(F\left(x^{2}+1\right)\right) $$

Short answer

The derivative is \( F'(x^2 + 1) \cdot 2x \).

Step-by-step solution

Identify the Composition

The function inside the differentiation operator is a composition, specifically, it is \( F(g(x)) \), where \( F(u) \) is the outer function and \( g(x) = x^2 + 1 \) is the inner function. We need to find the derivative of this composition.

Recall the Chain Rule

The Chain Rule is used to differentiate compositions of functions. It states that the derivative of \( F(g(x)) \) with respect to \( x \) is \( F'(g(x)) \cdot g'(x) \).

Differentiate the Outer Function

Differentiate the outer function \( F(u) \) with respect to \( u \). This gives us \( F'(u) \). After this, substitute back in for the function \( g(x) \) to get \( F'(x^2 + 1) \).

Differentiate the Inner Function

Differentiate the inner function \( g(x) = x^2 + 1 \) with respect to \( x \). This derivative is \( 2x \).

Combine Using the Chain Rule

Apply the Chain Rule to combine the derivatives found in Steps 3 and 4: \( D_{x}(F(x^2+1)) = F'(x^2 + 1) \cdot 2x \).

Key concepts

Differentiation

Differentiation is a central concept in calculus that involves finding the rate at which a function is changing at any given point. In simple terms, it helps us determine the slope of the tangent line to the function at a particular point. This is achieved by computing the derivative of the function.

In the context of our original exercise, differentiation is the operation we perform to find how the function \( F(x^2+1) \) changes as \( x \) changes. This involves using differentiation rules such as the Chain Rule, which allows us to handle composite functions effectively.

Understanding differentiation involves familiarizing yourself with several basic rules:

• The Power Rule: if \( y = x^n \), then \( \frac{dy}{dx} = nx^{n-1} \)
• The Product Rule: for two functions \( u(x) \) and \( v(x) \), \( (uv)' = u'v + uv' \)
• The Quotient Rule: for two functions \( u(x) \) and \( v(x) \), \( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \)
• The Chain Rule: which is key in our exercise, used for differentiating compositions of functions.

Function Composition

In calculus, function composition involves combining two functions into a single function. This occurs when the output of one function becomes the input of another, creating a composite function. For example, the function \( F(g(x)) \) is formed by composing functions \( F \) and \( g \).

In our exercise, \( F(g(x)) \) represents \( F(x^2 + 1) \). The function \( g(x) = x^2 + 1 \) is evaluated first, followed by the application of \( F \) to this result. Here, the concept of composition is critical because it requires a specific differentiation strategy, namely the Chain Rule.

Compositions are important for:

• Transforming and simplifying complex mathematical expressions.
• Developing intricate models involving multiple mathematical processes.
• Analyzing systems in engineering and scientific applications.

Understanding function composition allows us to unravel complex systems and apply calculus techniques effectively, such as finding derivatives.

Calculus Problem Solving

Calculus problem solving involves applying a variety of techniques and strategies to tackle mathematical challenges relating to rates of change and areas. It requires a methodical approach to understand which rules and theorems apply in a given situation.

In solving our original exercise, we use:

• Identification: Recognize that the function is a composition, \( F(g(x)) = F(x^2 + 1) \).
• Logical Application: Apply the Chain Rule, which is specially designed to differentiate composite functions.
• Precision: Carefully compute each part of the derivative by focusing on both the outer function \( F(u) \) and inner function \( g(x) \).

To excel in calculus problem solving:

• Practice recognizing different function types like composites and products.
• Refine your skills in various differentiation techniques.
• Use logical reasoning to decide which rule to apply first in complex situations.
• Verify solutions to ensure they align with the original problem context and are mathematically sound.