Problem 3 - Calculus Early Transcendentals

Question

Find \(f^{\prime}(x)\) $$ f(x)=-4 x^{2} \cos x $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Function Components

The given function is a product of two functions: \(-4x^2\) and \(\cos x\). Label these as \(u(x) = -4x^2\) and \(v(x) = \cos x\).

Step 2: Apply Product Rule for Derivatives

To find the derivative of a product of two functions, \(f(x) = u(x)v(x)\), use the product rule: \[ f'(x) = u'(x)v(x) + u(x)v'(x) \]

Step 3: Differentiate Each Function Component

First, find the derivative of \(u(x) = -4x^2\). The derivative is:\(u'(x) = \frac{d}{dx}(-4x^2) = -8x\).Now, find the derivative of \(v(x) = \cos x\): The derivative is:\(v'(x) = \frac{d}{dx}(\cos x) = -\sin x\).

Step 4: Substitute into the Product Rule Formula

Substitute the derivatives into the product rule formula:\[ f'(x) = u'(x)v(x) + u(x)v'(x) \]Replacing \(u'(x)\) with \(-8x\), \(v(x)\) with \(\cos x\), \(u(x)\) with \(-4x^2\), and \(v'(x)\) with \(-\sin x\), we get:\[ f'(x) = (-8x)(\cos x) + (-4x^2)(-\sin x) \]

Step 5: Simplify the Expression

Simplify the derivative expression:\[ f'(x) = -8x \cos x + 4x^2 \sin x \]

Key Concepts

Product Rule

The product rule is a powerful tool in calculus for finding the derivative of products of two or more functions. Specifically, if you have two functions, let's call them \(u(x)\) and \(v(x)\), and you want the derivative of their product, you would use the product rule. The formula for the product rule is:\[ f'(x) = u'(x)v(x) + u(x)v'(x) \]Here's what this means:

• Take the derivative of the first function \(u(x)\), which we denote as \(u'(x)\).
• Multiply it by the second function \(v(x)\) as it is (not differentiated).
• Then take the derivative of the second function \(v(x)\), noted as \(v'(x)\).
• Multiply it by the first function \(u(x)\) in its original form.

Finally, you add these two results together to get the final derivative of the product. This rule is essential when working with functions that cannot easily be separated or simplified further. Practicing applying this rule helps in solidifying the technique in various calculus problems.

Trigonometric Functions

Trigonometric functions, like \(\cos x\) and \(\sin x\), are fundamental in calculus. They are periodic functions, meaning they repeat their values in regular intervals. Because of this behavior, their derivatives also follow distinct patterns.When differentiating trigonometric functions, it's crucial to remember the following derivatives:

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

Understanding these patterns allows you to differentiate and work with these functions effectively in various calculus problems. Having these derivatives at your fingertips makes calculations faster and ensures accuracy.

Differentiation Techniques

Differentiation techniques are methods used to find the derivative of a function, which represents the rate of change. The goal is to understand how a small change in one variable affects another. Techniques such as the chain rule, product rule, and quotient rule are common tools in this quest.For differentiating polynomials, like \(-4x^2\), you apply the power rule:

• Multiply by the exponent \(n\).
• Reduce the exponent by one.

This technique simplifies to: if \(u(x) = ax^n\), then \(u'(x) = nax^{n-1}\). Applying this to \(-4x^2\), its derivative becomes \(-8x\).Using and combining these techniques effectively allows you to tackle complex differentiation problems by breaking them into manageable steps. The key is to practice each rule and recognize when they apply.