Proving the Derivative of the Sine Function: A Comprehensive Guide

The derivative of the sine function, sin(x), is a fundamental concept in calculus with numerous applications in various scientific fields. In this article, we will walk through a detailed proof that the derivative of sin(x) is cos(x) using the definition of the derivative and properties of limits. By the end, you will have a deep understanding of how this key result is derived.

Step 1: Definition of the Derivative

The derivative of a function fx at a point x is defined as:

fx limh to 0 *frac{fx h - fx}{h}

For the function fx sin x, the expression for the derivative is:

*frac{d}{dx} sin x limh to 0 *frac{sin(x h) - sin x}{h}

Step 2: Using the Sine Addition Formula

The sine addition formula states that:

sin(x h) sin x cos h cos x sin h

Substituting this into our limit expression:

*frac{d}{dx} sin x limh to 0 *frac{sin x cos h cos x sin h - sin x}{h}

Step 3: Simplifying the Expression

Let's rearrange the expression inside the limit:

*frac{d}{dx} sin x limh to 0 *frac{sin x cos h - 1 cos x sin h}{h}

This simplifies to:

*frac{d}{dx} sin x limh to 0 *left( *frac{sin x cos h - 1}{h} *frac{cos x sin h}{h} *right)

Step 4: Splitting the Limit

Splitting the limit into two parts, we get:

*frac{d}{dx} sin x limh to 0 *frac{sin x cos h - 1}{h} limh to 0 *frac{cos x sin h}{h}

Step 5: Evaluating Each Limit

Limit of the First Term: We know that the limit:

limh to 0 *frac{cos h - 1}{h} 0

Therefore:

limh to 0 *frac{sin x cos h - 1}{h} sin x * 0 0

Limit of the Second Term: We use the fact that:

limh to 0 *frac{sin h}{h} 1

Thus:

limh to 0 *frac{cos x sin h}{h} cos x * 1 cos x

Step 6: Combining the Results

Putting it all together, we find:

*frac{d}{dx} sin x 0 cos x cos x

Conclusion

Therefore, we have proved that:

*frac{d}{dx} sin x cos x

This proof demonstrates the fundamental nature of the derivative of the sine function and its relationship with the cosine function.

Additional Insight: Using the Limit Definition (Alternative Approach)

Another approach using the limit definition is as follows:

limh to 0 *frac{sin(x h) - sin x - h}{2h}

By applying the sine addition formula:

sin(x h) sin x cos h cos x sin h

We substitute and simplify:

limh to 0 *frac{sin x cos h cos x sin h - sin x - h}{2h} *frac{sin x cos h - 1 cos x sin h}{2h}

This breaks down to:

limh to 0 *frac{2 cos x sin h}{2h}

Which simplifies further to:

cos x * limh to 0 *frac{sin h}{h} cos x * 1 cos x

The smaller h gets, the quantity approaches 1. Therefore, the derivative is:

cos x

Example Application: Integral Calculus

The derivative of the sine function has wide-ranging applications in integral calculus. For example, if we want to evaluate an integral involving sine, such as:

∫ sin(x) dx

Knowing that:

∫ sin(x) dx -cos(x) C, where C is the constant of integration

We can use the knowledge of the derivative of sine to simplify and solve various integration problems.