Problem 23 - Calculus - AP Edition

Question

Find \(f^{\prime}(x)\) $$f(x)=\sqrt{\cos (5 x)}$$

Step-by-Step Solution

Answer

\( f'(x) = \frac{-5\sin(5x)}{2\sqrt{\cos(5x)}} \)

Step 1: Identify the Outer Function

The given function is \( f(x) = \sqrt{\cos(5x)} \). Here, the outer function is a square root, which can be expressed as an exponent, \( f(x) = (\cos(5x))^{1/2} \).

Step 2: Differentiate the Outer Function

To differentiate the function \( (\cos(5x))^{1/2} \), apply the power rule. The derivative of \( u^{1/2} \) with respect to \( u \) is \( \frac{1}{2}u^{-1/2} \). Therefore, the derivative of the outer function with respect to \( \cos(5x) \) is \( \frac{1}{2} (\cos(5x))^{-1/2} \).

Step 3: Identify the Inner Function

The inner function is \( \cos(5x) \). We will need to differentiate this as part of the chain rule.

Step 4: Differentiate the Inner Function

The derivative of \( \cos(5x) \) with respect to \( x \) is found using the chain rule. The derivative of \( \cos(u) \) is \(-\sin(u)\), then apply the chain rule to get the derivative, \( -\sin(5x) \cdot 5 = -5\sin(5x) \).

Step 5: Apply the Chain Rule

The chain rule states that the derivative of a composite function \( f(g(x)) \) is \( f'(g(x))g'(x) \). Combine the derivative of the outer function from Step 2 with the derivative of the inner function from Step 4: \[ f'(x) = \frac{1}{2}(\cos(5x))^{-1/2} \cdot (-5\sin(5x)) \].

Step 6: Simplify the Expression

Simplify the expression: \[ f'(x) = \frac{-5\sin(5x)}{2\sqrt{\cos(5x)}} \].

Key Concepts

Chain Rule in Differentiation

The chain rule is a key concept when dealing with functions composed of multiple layers, or composite functions. Essentially, it helps us differentiate a function that is the composition of two or more functions. To put it simply, if you have a function of the form \(f(g(x))\), the chain rule tells you how to find its derivative.

• First, you differentiate the outer function with respect to the inner function;
• Then, you multiply that by the derivative of the inner function.

This way, the chain rule works by "unwrapping" each layer, one step at a time. It emphasizes the relationship between the outer and inner functions, requiring you to pay attention to each layer's derivative.In our example, the chain rule was crucial for differentiating \( f(x) = \sqrt{\cos(5x)} \). First, we had to differentiate the square root layer (the outer function) and then the \( \cos(5x) \) component (the inner function). Finally, we combined these results to obtain the derivative.

The Power Rule and Its Application

The power rule is one of the most straightforward rules in calculus for differentiation. It states that for any function of the form \( u^n \), where \( u \) is a differentiable function of \( x \) and \( n \) is a real number, the derivative is \( nu^{n-1} \). This rule simplifies the process of finding derivatives, allowing for quick manipulations of expressions raised to a power.In the original exercise, we transformed the square root into an exponent when we rewrote the function as \( (\cos(5x))^{1/2} \). Applying the power rule, the derivative of \( u^{1/2} \) is \( \frac{1}{2}u^{-1/2} \). Here, \( u \) represents \( \cos(5x) \), and by differentiating, we treat it as a standalone component before addressing the inner chain with the chain rule.

Composite Functions Explained

Composite functions are functions made by combining two or more functions. More precisely, if you have functions \( f(x) \) and \( g(x) \), a composite function \( f(g(x)) \) means that you input \( g(x) \) into \( f(x) \) instead of \( x \).

• To simplify computations, identify and separate the inner and outer functions.
• Understand that part of solving composite functions involves using the chain rule to tackle each component distinctly.

In the given problem of differentiating \( f(x) = \sqrt{\cos(5x)} \), the composite nature of the function is evident. \( \sqrt{x} \) is the outer function, applied to the result of \( \cos(5x) \), the inner function. Recognizing this composite structure is crucial since it directs how we apply the chain rule effectively. It allows us to break down the process into manageable steps, focusing on differentiating each part correctly.