Problem 72 - Calculus Early Transcendentals

Question

Use the derivative formula for sin \(x\) and the identity $$ \cos x=\sin \left(\frac{\pi}{2}-x\right) $$ to obtain the derivative formula for cos x.

Step-by-Step Solution

Answer

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

Step 1: Understand the Derivative of sin(x)

The derivative of the function \( \sin(x) \) with respect to \( x \) is known and given by \( \frac{d}{dx} \sin(x) = \cos(x) \). We will use this information along with trigonometric identities to find the derivative of \( \cos(x) \).

Step 2: Use the Cosine Identity

We are given the trigonometric identity \( \cos(x) = \sin\left( \frac{\pi}{2} - x \right) \). This identity allows us to express \( \cos(x) \) in terms of the \( \sin(x) \) function, which we have already worked with in terms of derivatives.

Step 3: Differentiate Using the Chain Rule

To find the derivative of \( \cos(x) \), take the derivative of \( \sin\left( \frac{\pi}{2} - x \right) \). Using the chain rule, we differentiate the outer function and multiply it by the derivative of the inner function:\[ \frac{d}{dx} \sin\left( \frac{\pi}{2} - x \right) = \cos\left( \frac{\pi}{2} - x \right) \cdot \left(-1\right) \]This simplification yields:\[ -\cos\left( \frac{\pi}{2} - x \right) \]

Step 4: Simplify Using the Identity Again

Recall that \( \cos\left( \frac{\pi}{2} - x \right) = \sin(x) \). Substituting this into the expression, we get:\[ -\cos\left( \frac{\pi}{2} - x \right) = -\sin(x) \]

Step 5: Conclusion of Derivative

Therefore, the derivative of \( \cos(x) \) with respect to \( x \) is:\[ \frac{d}{dx} \cos(x) = -\sin(x) \]

Key Concepts

Trigonometric Identities

Trigonometric identities are crucial in simplifying and manipulating expressions involving trigonometric functions. They play a significant role in calculus, especially when we are working to find derivatives.
For the function \( \cos(x) \), a powerful identity is \(\cos(x) = \sin\left( \frac{\pi}{2} - x \right) \).
This identity expresses cosine in terms of sine, allowing us to use our understanding of sine derivatives to find the cosine derivative.

• **Key Identity**: \( \cos(x) = \sin\left( \frac{\pi}{2} - x \right) \)
• **Purpose**: Facilitates the differentiation of cosine by relating it to sine functions.
• **Application**: Used extensively in calculus to simplify derivatives involving trigonometric functions.

The identity shows a deep interconnection between sine and cosine, where shifting angles can transform one function into another. Understanding this relationship allows us to tackle complex differential equations more efficiently.

Chain Rule

The chain rule is an essential differentiation technique in calculus. It is used when dealing with composite functions—functions nested within each other.
The rule states that to differentiate a composite function like \( f(g(x)) \), you first differentiate \( f \) with respect to \( g \), and then multiply by the derivative of \( g \) with respect to \( x \).
For example, to differentiate \( \sin\left( \frac{\pi}{2} - x \right) \) to find the derivative of \( \cos(x) \), we use the chain rule:

• **Differentiate the outer function**: As \( \sin(u) \), where \( u = \frac{\pi}{2} - x \).
• **Multiply by the derivative of the inner function**: Derivative of \( u \) with respect to \( x \), which is \( -1 \).

This results in the expression \( \cos\left( \frac{\pi}{2} - x \right) \cdot (-1) \). Applying the chain rule simplifies finding the derivative of more complicated nested functions, streamlining the process significantly.

Differentiation Techniques

Differentiation techniques are methods used to find the derivative of functions, which represent the rate at which a quantity changes. In this context, understanding these techniques is fundamental.
Derivatives are essential in predicting motion and change, particularly when evaluating trigonometric functions like sine and cosine.
Here are some key differentiation techniques relevant for this exercise:

• **Basic Derivatives**: Knowing that \( \frac{d}{dx} \sin(x) = \cos(x) \) and that the derivative of a constant is 0.
• **Chain Rule**: As seen before, used for composite functions.

Applying these techniques, especially in combination, allows us to tackle diverse derivative problems effectively. For instance, the process of obtaining the derivative \( \frac{d}{dx} \cos(x) = -\sin(x) \) arises from utilizing the basic derivative of sine along with the chain rule, leading to a comprehensive and simple solution to a potentially complex problem.