Problem 13 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Differentiate the given function. $$f(x)=\cos ^{2} x$$

Step-by-Step Solution

Answer

The derivative is \( -2 \text{cos }x \text{sin }x \)

Step 1: Rewrite the function in an alternative form

Rewrite the given function using the chain rule format as follows: \[ f(x) = (\text{cos }x)^2 \]

Step 2: Apply the Chain Rule

To differentiate the function, use the chain rule. According to the chain rule, \[ \frac{d}{dx} [u(x)]^n = n [u(x)]^{n-1} \frac{d}{dx} u(x) \]In this case, \(u(x) = \text{cos }x\) and \(n = 2\).Therefore, let \(f(x) = [\text{cos }x]^2\)

Step 3: Differentiate the outer function

Differentiate the outer function which is \((u(x))^2\). According to the chain rule, the derivative is given by:\[ 2 [\text{cos }x]^{2-1} \frac{d}{dx} [\text{cos }x] \]This simplifies to:\[ 2 \text{cos }x (-\text{sin }x) \]

Step 4: Simplify the expression

Simplify the final expression for the derivative now: \[ -2 \text{cos }x \text{sin }x \]The derivative of the function \(f(x)=\text{cos }^2 x\) is \(-2 \text{cos }x \text{sin }x\)

Key Concepts

chain rule

The chain rule is a crucial concept in calculus for differentiating composite functions. It tells us how to find the derivative of functions made by combining simpler functions. To use the chain rule, we follow this formula: \ \[ \frac{d}{dx} [u(x)]^n = n [u(x)]^{n-1} \frac{d}{dx} u(x) \] \ This formula essentially means we first differentiate the outer function while keeping the inner function unchanged. Next, we multiply by the derivative of the inner function. This step-by-step approach allows us to handle even complex function combinations smoothly.

derivative

Derivatives measure how a function changes as its input changes. In other words, the derivative is the rate of change or slope of the function at a particular point. The notation commonly used for derivatives is \ \( f'(x) \) or \ \( \frac{df}{dx} \). \ Calculating derivatives helps us understand the behavior of functions like their trends, maxima and minima. For a function like \ \( f(x) = \text{cos}^2 x \), applying the derivative reveals the rate at which \ \( \text{cos}^2 x \) changes with respect to \ \( x \). \ It's essential to know basic differentiation rules, such as the power rule, product rule, quotient rule, and chain rule, to derive functions efficiently.

trigonometric functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. Essential trigonometric functions include sine (\( \text{sin} \)), cosine (\( \text{cos} \)), and tangent (\( \text{tan} \)). These functions are cyclic and repeat their values in a regular pattern. For differentiation, trig functions have specific derivatives: \

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• \( \frac{d}{dx} \text{sin } x = \text{cos } x \)
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• \( \frac{d}{dx} \text{cos } x = - \text{sin } x \)
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• \( \frac{d}{dx} \text{tan } x = \text{sec}^2 x \)
\

These derivatives can be combined using the chain rule, like in this example \ \( f(x) = \text{cos}^2 x \), where we differentiated \ \( \text{cos } x \). Understanding how trigonometric functions are derived is essential for solving problems involving periodic behavior and waveforms.

product rule

The product rule is used when differentiating the product of two functions. It's vital in many cases, especially in physics and engineering. The product rule states: \ \[ \frac{d}{dx} [u(x) \times v(x)] = u(x) \frac{d}{dx} v(x) + v(x) \frac{d}{dx} u(x) \]\ This means to take the derivative of the first function and multiply it by the second function, then add the product of the derivative of the second function and the first function. Although the given problem does not directly use the product rule, it's helpful to recognize that understanding multiple rules like the chain rule and product rule enables tackling more complicated derivatives accurately.