Problem 21 - Calculus for Biology and Medicine

Question

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=4 \cos x^{2}-2 \cos ^{2} x $$

Step-by-Step Solution

Answer

The derivative of the function is \(f'(x) = -8x \sin(x^2) + 4 \cos(x) \sin(x)\).

Step 1: Identify the function components

The function given is composed of two parts: \(4 \cos(x^2)\) and \(-2 \cos^2(x)\). Each part needs to be differentiated separately with respect to \(x\).

Step 2: Differentiate the first component

The first component of the function is \(4 \cos(x^2)\). To differentiate it, apply the chain rule. The derivative of \(\cos(u)\) with respect to \(u\) is \(-\sin(u)\), and the derivative of \(x^2\) with respect to \(x\) is \(2x\). So, the derivative is:\[-4 \cdot \sin(x^2) \cdot 2x = -8x \sin(x^2)\].

Step 3: Differentiate the second component

The second component of the function is \(-2 \cos^2(x)\). Use the chain rule for \(\cos^2(x)\), noting that it's \((\cos(x))^2\). Differentiate using the chain rule:The outer function is \(u^2\), whose derivative is \(2u\), and the inner function is \(\cos(x)\), whose derivative is \(-\sin(x)\). Apply the product of these derivatives:\[-2 \cdot 2 \cos(x) \cdot (-\sin(x)) = 4 \cos(x) \sin(x)\].

Step 4: Combine the derivatives

Now, add the derivatives of both components to find \(f'(x)\):\[f'(x) = -8x \sin(x^2) + 4 \cos(x) \sin(x)\].

Key Concepts

Chain Rule

The chain rule is an essential tool in calculus, especially when dealing with composite functions. It helps us differentiate complex functions by breaking them down into simpler parts.

Here's how it works:

• Identify the outer and inner functions. The outer function acts on the result of the inner function.
• Differentiate the outer function with respect to the inner function.
• Differentiate the inner function with respect to the independent variable.
• Multiply the derivatives together.

In our problem, the chain rule is applied to both components:

• For the first part, \(4 \cos(x^2)\), \(\cos(u)\) is the outer function and \(x^2\) is the inner function. By differentiating, we find it as \(-8x \sin(x^2)\).
• For the second part, \(-2 \cos^2(x)\), we treat it as \((\cos(x))^2\). The outer function is \(u^2\) while the inner function is \(\cos(x)\). Differentiating gives us \(4 \cos(x) \sin(x)\).

Understanding how to apply the chain rule will boost your ability to approach complex differentiation problems efficiently.

Trigonometric Functions

Trigonometric functions like sine and cosine often appear in calculus problems. These functions are periodic and have specific derivative forms that are crucial to know.

Here's a quick refresher:

• The derivative of \(\sin(x)\) is \(-\cos(x)\).
• The derivative of \(\cos(x)\) is \(-\sin(x)\).

In the problem given, both the \(-\sin(x^2)\) and \(\cos(x) \sin(x)\) terms are derived from using these concepts.

• Recognizing these basic derivatives will help you quickly find the rate of change of more complex trigonometric expressions.
• These functions help describe cyclic phenomena, an essential part of understanding their role in modeling real-world scenarios like waves.

Differentiation Techniques

Differentiation is a procedure used to find the derivative of a function, telling us the function's rate of change. Several techniques are used, depending on the function's complexity.

• Basic differentiation: Use this for simple functions like polynomials (e.g., the derivative of \(x^n\) is \(nx^{n-1}\)).
• Chain rule: Essential when dealing with composite functions, as covered previously.
• Product rule: Use when you need to differentiate products of two functions (e.g., if \(y = uv\), then \(y' = u'v + uv'\)).

In our problem, we used the chain rule for the composite functions and applied basic trigonometric derivatives.

By combining these techniques, you can handle a wide range of problems, ensuring each component of the expression is accurately differentiated. Mastering these techniques solidifies your ability to approach calculus problems with confidence.