Problem 27 - Calculus - AP Edition

Question

Find \(d y / d x\) $$y=x^{3} \sin ^{2}(5 x)$$

Step-by-Step Solution

Answer

The derivative is \(3x^2 \sin^2(5x) + 5x^3 \sin(10x)\).

Step 1: Identify the Function Type

The function given is a product of two functions, namely: \(f(x) = x^3\) and \(g(x) = \sin^2(5x)\). This indicates that we should use the product rule to differentiate.

Step 2: Review the Product Rule

The product rule for differentiation states: \(\frac{d}{dx}(u \cdot v) = u' \cdot v + u \cdot v'\), where \(u\) and \(v\) are functions of \(x\).

Step 3: Differentiate x^3

Let \(u = x^3\). The derivative \(u'\) is \(3x^2\).

Step 4: Differentiate sin^2(5x)

Let \(v = \sin^2(5x)\). Use the chain rule and the fact that \(\frac{d}{dy}(y^2) = 2y\) and \(\frac{d}{dx}(\sin(5x)) = 5 \cos(5x)\). Differentiating \(v\), we get: \(v' = 2 \sin(5x) \cdot 5 \cos(5x) = 10 \sin(5x) \cos(5x)\).

Step 5: Apply the Product Rule

Using the product rule: \(\frac{d}{dx}(x^3 \sin^2(5x)) = 3x^2 \cdot \sin^2(5x) + x^3 \cdot 10 \sin(5x) \cos(5x)\).

Step 6: Simplify the Expression

The expression simplifies to: \(3x^2 \sin^2(5x) + 10x^3 \sin(5x) \cos(5x)\). Recognize \(2 \sin(5x) \cos(5x)\) as \(\sin(10x)\), from the double-angle identity. Thus, the derivative is \(3x^2 \sin^2(5x) + 5x^3 \sin(10x)\).

Key Concepts

Product Rule

The product rule is an essential technique in calculus differentiation. It helps us differentiate the product of two functions. If you have two functions multiplied together, like in our exercise where we have \(f(x) = x^3\) and \(g(x) = \sin^2(5x)\), the product rule comes into play. The product rule formula is \(\frac{d}{dx}(u \cdot v) = u' \cdot v + u \cdot v'\). This means:

• Take the derivative of the first function \(u\), and multiply it by the second function \(v\).
• Then, take the first function \(u\) and multiply it by the derivative of the second function \(v\).
• Add the two resulting expressions together.

In our exercise, we first found the derivative of \(x^3\), which is \(3x^2\). Then, applying the product rule gives the result by combining it with the derivative of \(\sin^2(5x)\). The product rule is perfect for functions expressed as products.

Chain Rule

The chain rule is a method used in calculus differentiation to differentiate composite functions. A composite function is one where a function is inside another function. In the exercise, \(\sin^2(5x)\) is actually \((\sin(5x))^2\), which shows a function inside another function.

• Think of it as an outside function and an inside function. Here, the outer function is \(y^2\) while the inner function is \(\sin(5x)\).
• The chain rule formula states: \(\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)\).
• Start by differentiating the outer function based on the inner one; here \(2 \cdot \sin(5x)\).
• Multiply by the derivative of the inner function. Here, the derivative of \(\sin(5x)\) is \(5\cos(5x)\).

So, the derivative of \(\sin^2(5x)\) results in \(10\sin(5x)\cos(5x)\). Mastering the chain rule means you can handle even complex differentiations.

Trigonometric Identities

Trigonometric identities are formulas involving trigonometric functions. They are useful to simplify expressions before or after differentiation. In the exercise, the trigonometric identity used was the double-angle identity.

• Recognize that \(2\sin(a)\cos(a) = \sin(2a)\). This is a common identity that simplifies expressions with sines and cosines.
• In our solution, the expression \(10\sin(5x)\cos(5x)\) simplifies using the identity, which transforms into \(5\sin(10x)\).
• This simplification reduces complexity and often provides a more concise answer.

Understanding and applying trigonometric identities can make a big difference when working through derivatives, ensuring solutions are both accurate and efficient.