Question

Use the trigonometric identity \(\cos 2 x=2 \cos ^{2} x-1\) along with the Product Rule to find \(D_{x} \cos 2 x\).

Short answer

The derivative \(D_x \cos 2x\) is \(-2\sin 2x\).

Step-by-step solution

Understand the Problem

To solve this problem, we need to find the derivative of the function \( \cos 2x \) using the given trigonometric identity \( \cos 2x = 2 \cos^2 x - 1 \). This involves using the Product Rule for differentiation.

Differentiate Using the Trigonometric Identity

Using the identity, \( \cos 2x = 2 \cos^2 x - 1 \), take the derivative with respect to \( x \). This means we have to differentiate \( 2 \cos^2 x - 1 \). The derivative of a constant, \(-1\), is 0, so we only need to focus on differentiating \( 2 \cos^2 x \).

Apply the Product Rule

The expression \( 2 \cos^2 x \) can be seen as \( 2 \times (\cos x)(\cos x) \). Using the Product Rule, \((uv)' = u'v + uv'\), let \( u = \cos x \) and \( v = \cos x \). Thus, \( u' = -\sin x \) and \( v' = -\sin x \). Apply the Product Rule: \( \frac{d}{dx}(\cos x \cdot \cos x) = (-\sin x)(\cos x) + (\cos x)(-\sin x) = -2\cos x \sin x \). Multiply by 2 to account for the coefficient: \( 2(-2\cos x \sin x) = -4\cos x \sin x \).

Simplify the Derivative

From the previous step, we obtained \( -4\cos x \sin x \) as the derivative. Recognize that \( \cos 2x = \cos(x + x) \) which can be differentiated using the chain rule resulting in \(-2\sin 2x\). For consistency, notice that \(-4\cos x \sin x\) is equivalent by the identity \( 2\sin x \cos x = \sin 2x \): thus \(-4\cos x \sin x = -2 \sin 2x \).

Verify the Result

Since the derivative \( D_x \cos 2x = -2\sin 2x \) obtained in different approaches is consistent with known differentiation rules, the solution is verified to be correct. Therefore, the derivative using the trigonometric identity and product rule is \(-2 \sin 2x\).

Key concepts

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions to one another. They are useful in simplifying expressions and solving equations where trigonometric functions are involved. For example, in our exercise, the identity \( \cos 2x = 2 \cos^2 x - 1 \) was pivotal. This identity allows us to express \( \cos 2x \) in terms of \( \cos x \), making it easier to compute derivatives or integrate these expressions.

Key trigonometric identities also include the Pythagorean identities, such as \( \sin^2 x + \cos^2 x = 1 \), which are essential in many calculus problems. Additionally, angle sum and difference identities, like \( \cos(a + b) = \cos a \cos b - \sin a \sin b \), help break down more complex expressions into simpler parts. Knowing and applying these identities is crucial when working with trigonometric functions in calculus.

• Helps in simplifying complex trigonometric expressions
• Enables solving trigonometric equations efficiently
• Facilitates finding derivatives and integrals of trigonometric functions

Product Rule

The Product Rule is a fundamental tool in calculus for finding the derivative of a product of two functions. If you have two functions, \( u(x) \) and \( v(x) \), the Product Rule states that the derivative of their product is \( (uv)' = u'v + uv' \).

In our solution, this principle was applied to the function \( 2 \cos^2 x \), which can be seen as a product of \( \cos x \) and \( \cos x \) itself. To differentiate, we set \( u = \cos x \) and \( v = \cos x \), resulting in \( u' = -\sin x \) and \( v' = -\sin x \). Applying the Product Rule gives us the derivative as \( -2\cos x \sin x \), which was further multiplied by 2 to account for the coefficient correction, leading to \( -4\cos x \sin x \).

• Useful for differentiating products of functions
• Combines derivatives of individual functions through addition
• Essential in handling complex expressions involving products

Chain Rule

The Chain Rule allows us to find the derivative of composite functions, which are functions nested within each other. It is expressed as \( (f(g(x)))' = f'(g(x)) \cdot g'(x) \).

In differentiation, the Chain Rule is particularly useful when dealing with expressions involving trigonometrics composed with other functions, like \( \cos(2x) \). In this context, recognizing \( \cos(2x) \) as \( \cos(y) \) where \( y = 2x \), allows us to take the derivative as \( -\sin(y) \cdot 2 \), which simplifies to \(-2 \sin(2x) \).

• Vital for differentiating nested functions
• Involves taking the derivative of the outer function and multiplying by the derivative of the inner function
• Particularly useful for trigonometric functions composed with other expressions

Calculus

Calculus is a branch of mathematics focused on change and motion; it includes two primary operations: differentiation and integration. Differentiation concerns the rate at which things change, like slopes of curves or, in physics terms, velocity.

This exercise is an excellent example of applying differentiation within calculus, highlighting how fundamental tools like trigonometric identities, the Product Rule, and the Chain Rule work together. Understanding differentiation entails mastering how to find slopes of functions, where you learn rules like the Product and Chain Rules, essential for working with composite and product forms.

• Primarily deals with differentiation and integration
• Used to find rates of change and areas under curves
• Crucial in fields like physics, engineering, and economics