Differentiate Sinx From First Principles

Differentiating sin x from First Principles: A Deep Dive

Understanding how to differentiate trigonometric functions like sin x from first principles is crucial for building a solid foundation in calculus. This process, which involves using the limit definition of the derivative, allows us to derive the derivative without relying on pre-established formulas. This article will guide you through a detailed explanation, tackling the derivation step-by-step and addressing common questions and potential stumbling blocks along the way. We will explore the underlying mathematical principles and provide intuitive explanations to enhance your comprehension. By the end, you'll not only understand how to differentiate sin x, but also why the derivative takes the form it does.

Introduction: The Limit Definition of the Derivative

The foundation of differential calculus lies in the concept of the derivative. The derivative of a function at a specific point represents the instantaneous rate of change of that function at that point. For a function f(x), its derivative at a point x is defined using the limit:

f'(x) = lim (h→0) [(f(x + h) - f(x)) / h]

This limit represents the slope of the tangent line to the graph of f(x) at the point x. If this limit exists, the function is said to be differentiable at x. We will use this definition to derive the derivative of sin x.

Step-by-Step Derivation of the Derivative of sin x

Let's differentiate f(x) = sin x from first principles. We'll substitute f(x) = sin x into the limit definition of the derivative:

f'(x) = lim (h→0) [(sin(x + h) - sin(x)) / h]

Now, we need to simplify this expression using trigonometric identities. Recall the trigonometric sum formula:

sin(A + B) = sin A cos B + cos A sin B

Applying this to our expression, where A = x and B = h, we get:

f'(x) = lim (h→0) [(sin x cos h + cos x sin h - sin x) / h]

We can rearrange this expression to separate the terms:

f'(x) = lim (h→0) [(sin x (cos h - 1) + cos x sin h) / h]

Now, we can split the limit into two separate limits:

f'(x) = lim (h→0) [sin x (cos h - 1) / h] + lim (h→0) [cos x sin h / h]

Since sin x and cos x are constants with respect to h, we can move them outside the limits:

f'(x) = sin x * lim (h→0) [(cos h - 1) / h] + cos x * lim (h→0) [sin h / h]

Now, we need to evaluate these two limits. This is where a deeper understanding of limits and trigonometric functions comes into play.

Limit 1: lim (h→0) [(cos h - 1) / h]

This limit is notoriously tricky to evaluate directly. However, we can use L'Hôpital's rule (which we'll explain later) or consider the Taylor series expansion of cos h around h = 0:

cos h ≈ 1 - h²/2! + h⁴/4! - ...

Substituting this into the limit:

lim (h→0) [(cos h - 1) / h] ≈ lim (h→0) [(-h²/2! + h⁴/4! - ...) / h] = lim (h→0) [-h/2! + h³/4! - ...] = 0

Therefore, the first limit evaluates to 0.

Limit 2: lim (h→0) [sin h / h]

This is another important limit in calculus. While not immediately obvious, this limit is famously equal to 1. Again, we can use L'Hôpital's rule or consider the Taylor series expansion of sin h around h = 0:

sin h ≈ h - h³/3! + h⁵/5! - ...

Substituting this into the limit:

lim (h→0) [sin h / h] ≈ lim (h→0) [(h - h³/3! + h⁵/5! - ...) / h] = lim (h→0) [1 - h²/3! + h⁴/5! - ...] = 1

Therefore, the second limit evaluates to 1.

Combining the Limits:

Now, we can substitute the values of the limits back into our expression for f'(x):

f'(x) = sin x * 0 + cos x * 1 = cos x

Therefore, the derivative of sin x with respect to x is cos x.

A Deeper Look: L'Hôpital's Rule and Taylor Series Expansions

We touched upon L'Hôpital's rule and Taylor series expansions. Let's delve a little deeper into these powerful tools.

L'Hôpital's Rule:

L'Hôpital's rule is a powerful technique for evaluating limits of indeterminate forms (like 0/0 or ∞/∞). It states that if the limit of f(x)/g(x) as x approaches a is an indeterminate form, and if the limit of f'(x)/g'(x) exists, then:

lim (x→a) [f(x)/g(x)] = lim (x→a) [f'(x)/g'(x)]

In our case, both limits we encountered were of the form 0/0. Applying L'Hôpital's rule:

• For lim (h→0) [(cos h - 1) / h]: The derivative of the numerator is -sin h, and the derivative of the denominator is 1. Thus, the limit becomes lim (h→0) [-sin h / 1] = 0.

• For lim (h→0) [sin h / h]: The derivative of the numerator is cos h, and the derivative of the denominator is 1. Thus, the limit becomes lim (h→0) [cos h / 1] = 1.

Taylor Series Expansions:

Taylor series expansions provide a way to approximate a function as an infinite sum of terms involving its derivatives at a specific point. The Taylor series expansion of a function f(x) around a point a is given by:

f(x) = f(a) + f'(a)(x-a)/1! + f''(a)(x-a)²/2! + f'''(a)(x-a)³/3! + ...

Using the Taylor series expansions for sin x and cos x around x = 0 significantly simplifies the evaluation of our limits. The expansions are readily available and provide a powerful alternative to L'Hôpital's rule.

Frequently Asked Questions (FAQ)

Q1: Why is the limit lim (h→0) [sin h / h] = 1?

A1: This is a fundamental limit in calculus. While it's not immediately obvious, it can be proven geometrically or using the squeeze theorem. Intuitively, as h approaches 0, the ratio of sin h to h approaches 1. The Taylor series expansion provides a straightforward algebraic proof.

Q2: Can we use L'Hôpital's rule for all limits?

A2: No. L'Hôpital's rule only applies to limits that are in indeterminate forms (0/0, ∞/∞, etc.). It cannot be applied to limits that are already defined or have a clear value.

Q3: What if I don't remember the trigonometric sum formula?

A3: If you forget the sum formula, you can derive it using the angle addition formulas for sine and cosine functions from the unit circle definition of trigonometric functions. Remembering the basic definitions is more helpful than memorizing every formula.

Q4: What are the practical applications of differentiating sin x?

A4: The derivative of sin x, cos x, is crucial in many fields, including physics (oscillatory motion, wave phenomena), engineering (signal processing), and computer graphics (modeling curves).

Q5: Are there other ways to differentiate sin x from first principles?

A5: Yes, while the method described above is widely used, alternative approaches might involve using geometric arguments or different trigonometric identities. However, the core principles remain the same - the limit definition of the derivative and an understanding of the behavior of trigonometric functions near zero.

Conclusion: Mastering the Fundamentals

Differentiating sin x from first principles might seem challenging initially, but by breaking down the process step-by-step and understanding the underlying concepts of limits, trigonometric identities, and potentially L'Hôpital's rule or Taylor expansions, it becomes much more manageable. This rigorous approach reinforces the fundamental principles of calculus and provides a deeper appreciation for the meaning of the derivative. The ability to derive the derivative of sin x from first principles demonstrates a solid understanding of calculus and its foundational concepts, crucial for further advancements in this field. Mastering this concept will provide you with a strong foundation for tackling more complex differentiation problems and further expanding your knowledge of calculus.