Problem 17 - University Calculus: Early Transcendentals

Question

Find \(d y / d x\). $$f(x)=x^{3} \sin x \cos x$$

Step-by-Step Solution

Answer

The derivative is \(3x^2 \sin x \cos x + x^3 (\cos^2 x - \sin^2 x)\).

Step 1: Identify the Function and Rule Needed

The function given is a product of two terms: \( x^3 \) and \( \sin x \cos x \). To find \( \frac{dy}{dx} \), we'll use the Product Rule, which states that \( \frac{d}{dx}(uv) = u'v + uv' \). Here, \( u = x^3 \) and \( v = \sin x \cos x \).

Step 2: Differentiate the First Term

Calculate the derivative of the first term \( u = x^3 \). Use the power rule which states \( \frac{d}{dx}(x^n) = nx^{n-1} \). Thus, \( u' = \frac{d}{dx}(x^3) = 3x^2 \).

Step 3: Differentiate the Second Term

Calculate the derivative of the second term \( v = \sin x \cos x \). Use the product rule again for this part since it is a product of two functions \( v = a(b) = \sin x (\cos x) \). Let \( a = \sin x \) and \( b = \cos x \).

Step 4: Apply the Product Rule to Second Term

Differentiate \( a = \sin x \) to get \( a' = \cos x \), and differentiate \( b = \cos x \) to get \( b' = -\sin x \). Apply the product rule: \( v' = a'b + ab' = (\cos x)(\cos x) + (\sin x)(-\sin x) = \cos^2 x - \sin^2 x \).

Step 5: Combine Using Product Rule for Entire Function

Now apply the Product Rule to the whole function: \( \frac{d}{dx}(uv) = u'v + uv' \), where \( u = x^3 \), \( u' = 3x^2 \), \( v = \sin x \cos x \), and \( v' = \cos^2 x - \sin^2 x \). This results in: \( 3x^2(\sin x \cos x) + x^3(\cos^2 x - \sin^2 x) \).

Step 6: Simplify the Expression

Combine and simplify the terms: \( 3x^2(\sin x \cos x) + x^3(\cos^2 x - \sin^2 x) \) by factoring if possible. No further simplification can combine these distinct terms, so this is the final derivative.

Key Concepts

Product Rule

The Product Rule is essential when you need to differentiate the product of two functions. The rule can be summarized as follows: if you have two functions, let's call them \(u\) and \(v\), their derivative is given by \((u'v + uv')\). Here’s a simple breakdown to help you understand this concept:

• Identify which parts of your function correspond to \(u\) and \(v\). In our exercise, \(u = x^3\) and \(v = \sin x \cos x\).
• Calculate the derivatives of each, which means you find \(u'\) and \(v'\).
• Substitute these derivatives back into the Product Rule formula to get the derivative of the whole function.

Using the Product Rule ensures that the complexities of multiplying two changing quantities are properly handled in differentiation. It’s an invaluable tool in calculus when dealing with such expressions.

Power Rule

The Power Rule is one of the simplest and most frequently used rules in differentiation, especially for polynomial functions. The rule states that for any real number \(n\), the derivative of \(x^n\) with respect to \(x\) is \(nx^{n-1}\). Let's delve into this with more detail:

• This rule applies to terms where \(x\) is raised to a constant power \(n\).
• To use it, multiply the entire term by the exponent \(n\) and then decrease the exponent by one.
• For the function \(x^3\), applying the Power Rule yields \(3x^{2}\) because you bring down the 3 (the exponent) and reduce the power by one.

Understanding the Power Rule makes it quicker and easier to tackle derivatives of polynomial functions without needing to use the more fundamental, limit-based definition of derivatives each time.

Trigonometric Derivatives

Differentiation of trigonometric functions is fundamental in calculus, as these functions are frequently encountered in various fields of science and engineering. In our exercise, you have \(\sin x\) and \(\cos x\) as parts of the term \(\sin x \cos x\). Here’s a concise guide:

• The derivative of \(\sin x\) is \(\cos x\).
• Conversely, the derivative of \(\cos x\) is \(-\sin x\).
• When dealing with a product of trigonometric functions, like \(\sin x \cos x\), you can apply the Product Rule for further differentiation.

Therefore, differentiating \(\sin x \cos x\), requires treating it as a product and applying the Product Rule to find \(v'\) as \(\cos^2 x - \sin^2 x\). Mastering these derivatives is crucial for analyzing oscillatory systems and wave phenomena.