Understanding the Derivative of the Function f(x) sin(x)/x

The function f(x) sin(x)/x is a classic example in calculus where the application of the quotient rule and limits play a crucial role. This article aims to provide a comprehensive guide to finding the derivative of this function, exploring the theory, derivations, and important considerations.

The Derivative Using the Quotient Rule

The quotient rule is a fundamental concept in calculus, used to find the derivative of a function that can be expressed as a ratio of two functions. The general form of the quotient rule is given by:

Quotient Rule: If f(x) g(x)/h(x), then the derivative f'(x) is given by:

f'(x) (g'(x)h(x) - g(x)h'(x))/[h(x)]2

Given the function f(x) sin(x)/x, let's break it down according to the quotient rule. Here, g(x) sin(x) and h(x) x. The derivatives of these functions are:

g'(x) cos(x)

h'(x) 1

Now, applying the quotient rule:

f'(x) [cos(x) * x - sin(x) * 1] / x2

f'(x) (x*cos(x) - sin(x))/x2

Another Approach: Using Limits and Series Expansion

For a more detailed understanding, consider the function defined as:

f(x) {1, if x 0

{sin(x)/x, if x ≠ 0}

In this case, the derivative at x 0 can be evaluated using limits. Using L'H?pital's rule or Taylor series expansion, we can show that:

limx→0 f(x) 0

By expanding sin(x) in a Taylor series around x 0, we get:

sin(x) x - x3/3! x5/5! - ...

Thus,

sin(x)/x 1 - x2/3! x4/5! - ...

Taking the derivative term by term, we obtain:

limx→0 f'(x) 0

Hence, the function can be defined as:

f(x) {0, if x 0

{(x*cos(x) - sin(x))/x2, if x ≠ 0}

Derivative via Implicit Differentiation

Another interesting approach is via implicit differentiation. Let:

y sin(x)/x

Then, taking the derivative of both sides with respect to x, we use the chain rule and quotient rule:

(dy/dx) (cos(x) * x - sin(x)) / x2

Conclusion

The derivative of the function f(x) sin(x)/x is found using the quotient rule, resulting in:

f'(x) (x*cos(x) - sin(x)) / x2

This article has explored the various methods to find the derivative of this function, emphasizing the importance of analytical and limit-based approaches in calculus. Understanding these concepts is fundamental for solving more complex problems in mathematics and related fields.