Problem 13 - Calculus

Question

Find the derivatives of the functions in \(1-26\). $$ x^{4} \cos x+x \cos ^{4} x $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 4x^3 \cos x - x^4 \sin x + \cos^4 x - 4x \cos^3 x \sin x \).

Step 1: Identify the functions for derivative

We are given the function \( f(x) = x^4 \cos x + x \cos^4 x \). This function is a sum of two separate terms: \( x^4 \cos x \) and \( x \cos^4 x \). We will find the derivative of each term individually.

Step 2: Differentiate the first term using product rule

For the first term \( x^4 \cos x \), apply the product rule. The product rule states that \( (uv)' = u'v + uv' \). Let \( u = x^4 \) and \( v = \cos x \). Now find each derivative:1. \( u' = 4x^3 \) 2. \( v' = -\sin x \)Using the product rule, we have \( (x^4 \cos x)' = 4x^3 \cos x - x^4 \sin x \).

Step 3: Differentiate the second term using chain and product rules

For the second term \( x \cos^4 x \), apply the product rule and chain rule. Let \( u = x \) and \( v = \cos^4 x \). Find the derivatives:1. \( u' = 1 \) 2. For \( v \), use the chain rule: \( v = (\cos x)^4 \Rightarrow v' = 4\cos^3 x \cdot (-\sin x) = -4\cos^3 x \sin x \).Using the product rule, \( (x \cos^4 x)' = 1 \cdot \cos^4 x - 4x \cos^3 x \sin x \).

Step 4: Combine the derivatives

Sum the derivatives found in steps 2 and 3: \[ f'(x) = (4x^3 \cos x - x^4 \sin x) + (\cos^4 x - 4x \cos^3 x \sin x) \].

Key Concepts

Product Rule

The product rule is a fundamental tool in calculus, specifically for differentiating products of two functions. It's essentially saying that the derivative of a product of two functions is not just the product of their derivatives. Instead, we calculate it as follows: if you have two functions, say \( u(x) \) and \( v(x) \), their product's derivative is given by \( (uv)' = u'v + uv' \).
This means you first find the derivative of the first function \( u(x) \), multiply it by the second function \( v(x) \), and then add the result to the derivative of the second function \( v(x) \) times the first function \( u(x) \).

• Find \( u' \), the derivative of \( u \).
• Multiply \( u' \) by \( v \).
• Find \( v' \), the derivative of \( v \).
• Multiply \( v' \) by \( u \).
• Sum the two results.

In this exercise, the product rule was used to differentiate both terms \( x^4 \cos x \) and \( x \cos^4 x \), highlighting the importance of identifying products correctly.

Chain Rule

The chain rule is another essential concept in calculus used for finding the derivative of a composition of functions. When you have a function nested inside another function, the chain rule is needed. For instance, if \( y = g(f(x)) \), the chain rule tells us that the derivative of \( y \) with respect to \( x \) is: \( y' = g'(f(x)) \cdot f'(x) \).
The rule fundamentally handles the change of the inner function and multiplies it by the change of the outer function. This comes into play when dealing with power functions that have a base of another function, such as \( \cos^4 x \).

• Identify the outer function \( g \) (such as raising to a power).
• Find the derivative of the outer function, keeping the inner function the same.
• Identify the inner function \( f \).
• Find the derivative of the inner function.
• Multiply these two derivatives together.

In this problem, we used the chain rule to correctly differentiate \( \cos^4 x \), focusing on the composition of the powers of a trigonometric function.

Trigonometric Functions

Trigonometric functions are a key element in calculus because they often appear in real-world applications involving cycles and waves. When dealing with derivatives, it's important to note the basic rules for differentiating these functions.

• The derivative of \( \sin x \) is \( \cos x \).
• The derivative of \( \cos x \) is \(-\sin x \).
• The derivative of \( \tan x \) is \( \sec^2 x \), though it was not needed in this exercise.

In the given equation \( f(x) = x^4 \cos x + x \cos^4 x \), you can see \( \cos x \) shows up in both terms. Thus, knowing how to differentiate \( \cos x \) as \(-\sin x \) is crucial. Furthermore, for \( \cos^4 x \), understanding that it's actually a power of a trigonometric function means you'll use the chain rule effectively. Recognizing these rules and when to apply them correctly is at the heart of comprehending and solving calculus problems.