Problem 234 - Calculus Volume 1

Question

Find \(\frac{d y}{d x}\) for each function. $$ y=x^{2} \cos ^{4} x $$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = 2x \cos^4 x - 4x^2 \cos^3 x \sin x \).

Step 1: Identify the Rule to Use

To find \( \frac{dy}{dx} \) for the function \( y = x^2 \cos^4 x \), we notice that this is a product of two functions: \( y = u \cdot v \), where \( u = x^2 \) and \( v = \cos^4 x \). We will use the product rule which states: \( \frac{d}{dx}(u \cdot v) = u' \cdot v + u \cdot v' \).

Step 2: Differentiate Each Function

First, differentiate \( u \, \text{and} \, v \). - For \( u = x^2 \), \( u' = \frac{d}{dx}(x^2) = 2x \).- For \( v = \cos^4 x \), apply the chain rule. Set \( w = \cos x \), then \( v = w^4 \) and use the chain rule: \( v' = 4w^3 \cdot (-\sin x) = -4 \cos^3 x \sin x \).

Step 3: Apply the Product Rule

Now, apply the product rule:\[ \frac{dy}{dx} = u' \cdot v + u \cdot v' \]Substitute the derivatives obtained:\[ \frac{dy}{dx} = (2x)(\cos^4 x) + (x^2)(-4 \cos^3 x \sin x) \]

Step 4: Simplify the Expression

Simplify the expression:\[ \frac{dy}{dx} = 2x \cos^4 x - 4x^2 \cos^3 x \sin x \]Factor each term if possible to further condense the expression if needed, but in this case, the expression is already simplified.

Key Concepts

Product Rule

The product rule is essential in calculus when dealing with the differentiation of functions that are multiplied together. This rule provides a way to differentiate a product of two functions. Suppose we have two functions, \( u(x) \) and \( v(x) \). The product rule states that the derivative of their product \( y = u(x) \cdot v(x) \) is given by:

• \( \frac{d}{dx}[u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) \)

This means that we need to calculate the derivative of each function separately, and then sum up the products of each function and the derivative of the other function.
In our original problem, we have \( y = x^2 \cdot \cos^4 x \), where \( x^2 \) is \( u \) and \( \cos^4 x \) is \( v \). By applying the product rule, we first calculated the derivatives \( u' = 2x \) and \( v' \) using another rule (Chain Rule), and then used these in the formula. This demonstrates the power of the product rule in handling complex derivatives.

Chain Rule

One of the most potent tools in the world of calculus is the chain rule. It is especially useful when dealing with composite functions, where one function is nested within another. Imagine a function \( y = f(g(x)) \). To find the derivative, known as \( \frac{dy}{dx} \), we use the chain rule:

• \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \)

This means we first take the derivative of the outer function \( f \) with respect to the inner function \( g \), and then multiply it by the derivative of the inner function \( g \) with respect to \( x \).
In our original exercise, the function \( v = \cos^4 x \) can be decomposed as \( w = \cos x \) leading to \( v = w^4 \). Using the chain rule, we differentiate \( w^4 \) to get \( 4w^3 \), multiply it by the derivative of \( \cos x \), which is \( -\sin x \). This gives us \( v' = -4 \cos^3 x \sin x \), showing the value and necessity of the chain rule in differentiating composite functions correctly.

Differentiation Techniques

Differentiation involves several techniques, each catering to specific types of functions. Key techniques include elementary derivatives, product rule, quotient rule, and more advanced ones like the chain rule.

• Basic Derivatives: Starting with the most simple derivative, such as \( \frac{d}{dx}(x^n) = nx^{n-1} \). This rule is foundational and frequently used in various contexts.
• Product Rule: As discussed earlier, the product rule allows us to differentiate products of two functions and is vital in numerous applications.
• Chain Rule: Another cornerstone, this rule deals with composite functions, enabling their differentiation in a systematic way.

By combining these techniques, we can tackle even the most complex functions. In our scenario, we used the basic derivative for \( x^2 \) and the product rule combined with the chain rule for \( \cos^4 x \). Understanding when and how to apply each technique is crucial for solving calculus problems effectively.