Problem 2 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Differentiate the function given. \(f(x)=3 \arctan (2 \sqrt{x})\)

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=3 \arctan (2 \sqrt{x})\) is \(\frac{3}{1+(2 \sqrt{x})^2} * \frac{1}{\sqrt{x}}\).

Step 1: Identifying the Inner and Outer Function

This expression can be considered a composition of the functions \(f(u) = 3 \arctan(u)\) and \(u = 2\sqrt{x}\). Here, \(f(u) = 3 \arctan(u)\) is the outer function and \(u = 2\sqrt{x}\) is the inner function.

Step 2: Compute Derivative of Outer Function

The derivative of the outer function \(f(u) = 3 \arctan(u)\) using the rule for the derivative of arctan is computed as follows: \(f'(u) = \frac{3}{1 + u^2}\).

Step 3: Compute Derivative of Inner Function

The derivative of the inner function \(u = 2\sqrt{x}\) is derived using the rule for the derivative of sqrt, as follows: \(g'(x)=2*\frac{1}{2*\sqrt{x}} = \frac{1}{\sqrt{x}}\).

Step 4: Applying the Chain Rule

Now the chain rule is applied, where \(dy/dx = dy/du * du/dx\). After substituting for \(\frac{dy}{du}\) and \(\frac{du}{dx}\), the derivative of the original function is found to be \(\frac{dy}{dx} = f'(u)*g'(x) = \frac{3}{1+(2 \sqrt {x})^2} * \frac{1}{\sqrt{x}}\).

Key Concepts

Derivative of Arctan

Understanding how to take the derivative of the arctangent function, or arctan, is crucial for tackling a variety of calculus problems. The arctan function is the inverse of the tangent function, and its derivative is a well-established result in calculus.
When you come across the function arctan, or \( \arctan(x) \), you'll use the formula \( \frac{d}{dx}(\arctan(x)) = \frac{1}{1+x^2} \) to find its derivative. This formula represents how quickly the angle of arctan changes with respect to changes in \( x \). The beautiful aspect of this derivative is that it doesn't matter whether \( x \) itself is a plain variable or a more complex expression; the formula applies equally well because of the chain rule.
In the case of our exercise, where the function is \( 3\arctan(2\sqrt{x}) \), we multiply the derivative of arctan by 3 in accordance with the constant multiple rule. This gives us the derivative of the outer function as \( \frac{3}{1+u^2} \) when considering \( u \) to be \( 2\sqrt{x} \).
It's important to approach this systematically, acknowledging the constant and the argument of the arctan separately, to ensure a correct application of the rules for differentiation.

Differentiating Composite Functions

When faced with a composite function such as \( f(x) = 3\arctan(2\sqrt{x}) \), understanding the mechanics of differentiating composite functions is key. These are functions nested within each other, like Russian dolls. The differentiation process for such composite or 'nested' functions is governed by the chain rule which connects the derivatives of the inner and outer functions.
In our exercise, the inner function, \( u = 2\sqrt{x} \), might seem tricky at first glance due to the square root. However, it simplifies to \( u' = \frac{1}{\sqrt{x}} \) when differentiated. This is your \( \frac{du}{dx} \) in the chain rule application.
Essentially, the chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. Mathematically put, for \( f(g(x)) \), it would be \( f'(g(x)) \cdot g'(x) \).

Why Chain Rule Works

Imagine you're observing how a change in \( x \) affects \( f(x) \), while \( x \) first runs through \( g(x) \). The chain rule captures this two-step process of change - first measuring the direct impact on \( g(x) \), and then understanding how that change impacts \( f(g(x)) \) subsequently. Without this rule, differentiating composite functions would be significantly more challenging, if not impossible in some cases.

Calculating Derivatives Step-by-Step

Approaching derivatives step-by-step is vital to avoid errors and to ensure a clear understanding of the differentiation process. The step-wise approach allows us to tackle complex problems by breaking them down into manageable pieces.
In our textbook exercise, we begin by identifying the inner function \( u = 2\sqrt{x} \) and the outer function \( f(u) = 3\arctan(u) \) and then proceed to differentiate them separately. By doing so, we unpack the problem and lay out each component clearly before applying the chain rule.
Each step is methodical:

• Identify Inner and Outer Functions: Recognize the composition of functions and label them accordingly.
• Differentiate Outer Function: Find the derivative of the outer function with respect to its input.
• Differentiate Inner Function: Find the derivative of the inner function with respect to \( x \).
• Apply Chain Rule: Multiply the derivative of the outer function by the derivative of the inner function to get the overall derivative.

This systematic process, as demonstrated in the given solution, ensures that each part of the composite function is accounted for and that the chain rule is correctly applied. Taking derivatives step-by-step is an invaluable strategy, offering clarity and accuracy especially when dealing with complicated expressions.