Problem 16 - The Calculus with Analytic Geometry

Question

Differentiate the given function by applying the theorems of this section. $$ g(x)=\left(4 x^{2}+3\right)^{2} $$

Step-by-Step Solution

Answer

The derivative is \(g'(x) = 16x(4x^2 + 3)\).

Step 1: Recognize the Outer Function

The given function is a composite function. Recognize that it is of the form \(f(u) = u^2\), where \(u = 4x^2 + 3\).

Step 2: Apply the Chain Rule

The Chain Rule states: \((f(g(x)))' = f'(g(x)) \cdot g'(x)\). Here, \g(x) = 4x^2 + 3\. Differentiate \(f(u) = u^2\) to get \(f'(u) = 2u\).

Step 3: Differentiate the Inner Function

Now, find the derivative of the inner function \(u = 4x^2 + 3\). The derivative is \(g'(x) = 8x\).

Step 4: Combine Results

After applying the Chain Rule, combine the differentiated parts: \(g'(x) = f'(u) \cdot g'(x) = 2u \cdot 8x\). Substitute \(u = 4x^2 + 3\) back in to get the final derivative.

Step 5: Simplify

Simplify the expression \(2(4x^2 + 3) \cdot 8x = 16x(4x^2 + 3)\). This is the derivative of the given function.

Key Concepts

Chain Rule

The chain rule is a fundamental rule in calculus for finding the derivative of composite functions. When you have a function composed of two or more functions, it is termed as a composite function. The chain rule states that to differentiate such a composite function, you must take the derivative of the outer function, keeping the inner function unchanged, and then multiply it by the derivative of the inner function. For example, if you have a function \(f(g(x))\), then the derivative is given by:

• \frac{d}{dx}[f(g(x))] = f'(g(x)) \times g'(x)

This approach helps in breaking down more complex differentiation problems into manageable steps, making it easier to solve them. It is important to recognize both the inner and outer functions in these problems to apply the chain rule correctly.

Composite Function

A composite function is created when one function is nested inside another. For instance, if you have two functions \(f(x)\) and \(g(x)\), then the composite function is denoted as \(f(g(x))\). In this case, \(g(x)\), the inner function, is first applied to \(x\), and then the outer function \(f(x)\) is applied to the result of \(g(x)\). For the example function \(g(x) = (4x^2 + 3)^2\), the inner function is \(u = 4x^2 + 3\), and the outer function is \(f(u) = u^2\). Recognizing these functions is crucial for applying the chain rule appropriately. Composite functions often appear in various mathematical contexts and understanding them is key in tackling complex differentiation problems.

Derivative

The derivative of a function measures how the function's output changes as its input changes. It tells you the slope of the tangent line to the function's graph at any given point. The differentiation process involves applying rules such as the chain rule to find this rate of change. In our exercise, we start with the function \(g(x) = (4x^2 + 3)^2\). The steps involve identifying the inner and outer functions, applying the chain rule, differentiating each part separately, and then combining the results. For example, the inner function \(u = 4x^2 + 3\) has a derivative of \(u' = 8x\), and the outer function \(f(u) = u^2\) has a derivative of \(f'(u) = 2u\). Combining these using the chain rule, we get the derivative of the entire function:

• 2(4x^2 + 3) \times 8x = 16x(4x^2 + 3)

This final expression represents the rate of change of the original function \(g(x)\).