Problem 192 - Calculus Volume 1

Question

Find \(\frac{d^{2} y}{d x^{2}}\) for the given functions. $$ y=\sin x \cos x $$

Step-by-Step Solution

Answer

The second derivative \( \frac{d^2y}{dx^2} = -2\sin(2x) \).

Step 1: Apply the Product Rule

The given function is a product of two functions: \( u = \sin x \) and \( v = \cos x \). The product rule for differentiation states: \( \frac{d}{dx}(uv) = u'v + uv' \). Differentiating \( u \) and \( v \), we get \( u' = \cos x \) and \( v' = -\sin x \). So, the first derivative is \( \frac{dy}{dx} = \cos x \cdot \cos x + \sin x \cdot (-\sin x) = \cos^2 x - \sin^2 x \).

Step 2: Simplify using Trigonometric Identity

Notice that \( \cos^2 x - \sin^2 x = \cos(2x) \), based on the double angle identity. Therefore, the first derivative \( \frac{dy}{dx} = \cos(2x) \).

Step 3: Differentiate using Chain Rule

Now differentiate \( \frac{dy}{dx} = \cos(2x) \) to find \( \frac{d^2y}{dx^2} \). Let \( w = 2x \), so using the chain rule: \( \frac{d}{dx} \cos(w) = -\sin(w) \times \frac{dw}{dx} \). Here, \( \frac{dw}{dx} = 2 \), thus \( \frac{d}{dx} \cos(2x) = -\sin(2x) \times 2 = -2\sin(2x) \). So, \( \frac{d^2y}{dx^2} = -2\sin(2x) \).

Key Concepts

Product Rule

The product rule is a fundamental tool in calculus used to differentiate functions that are the product of two other functions. Imagine you have a function that looks like this: \( y = u(x) \cdot v(x) \). The product rule helps you find the derivative of \( y \) by using the derivatives of \( u(x) \) and \( v(x) \). Instead of differentiating one part at a time and missing part of the function, the product rule formula ensures you capture all interactions:

• Formula: \( \frac{d}{dx}(uv) = u'v + uv' \)

There are a few steps to follow: - **Differentiate** the first function (\( u \)) and leave the second function (\( v \)) as is.- **Differentiate** the second function (\( v \)) and leave the first function (\( u \)) as is.- **Multiply** the differentiated function by the non-differentiated function each time, and **add** the results.
In our example, you differentiate \(u = \sin x\) as \(u' = \cos x\) and \(v = \cos x\) as \(v' = -\sin x\). The result, \(\frac{dy}{dx} = \cos^2 x - \sin^2 x\), is obtained by applying the product rule carefully. It's like filling in the pieces of a puzzle to get the complete picture.

Chain Rule

The chain rule provides a method for differentiating compositions of functions. Think of it like peeling an onion: you differentiate the outer layer first, then work your way inside. This is very useful when you have a function within another function, like in our example \( \cos(2x) \).
To apply the chain rule:

• Let \( w = g(x) \), making \( f(w) = f(g(x)) \).
• Differentiating \( f \) with respect to \( w \), get \( f'(w) \).
• Differentiating \( w \) with respect to \( x \), get \( g'(x) \).
• The chain rule formula is: \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \).

A quick tip is to always remember to multiply the derivative of the outer function by the derivative of the inner function. In our exercise, you set \(w = 2x\) and differentiate \(\cos(w)\) as \(-\sin(w)\), and use \(\frac{dw}{dx} = 2\). This gives you the second derivative: \(\frac{d^2y}{dx^2} = -2\sin(2x)\).

Trigonometric Identities

Trigonometric identities are conditions that hold true for all values of the involved variables. They are especially useful for simplifying expressions. In this exercise, recognizing such an identity simplifies the derivative immensely.
A popular identity is the double angle identity:

• \( \cos^2 x - \sin^2 x = \cos(2x) \)

These identities can transform complicated expressions into more manageable forms. In our solution, after applying the product rule, we simplify \(\cos^2 x - \sin^2 x\) into \(\cos(2x)\) using the double angle identity. This makes the subsequent differentiation steps much easier.
Getting familiar with these identities can be the key to solving many trigonometric differentiation problems, as they can bridge gaps between complex-looking derivatives and simple expressions.