Deriving the Derivative of sin(x)cos(x) from First Principles

The function y sin(x)cos(x) can be derived using various methods. In this article, we will explore both the product rule and the use of trigonometric identities, providing a comprehensive understanding of how to find its derivative. Additionally, we will use the first principles of calculus to derive the solution.

Using the Product Rule

The product rule is a fundamental method in differentiation used when a function is the product of two other functions. For functions f(x) sin(x) and g(x) cos(x), the product rule states:

derivative of [f(x)g(x)]  f'(x)g(x)   f(x)g'(x)

Applying this to our function:

f(x) sin(x) rarr; f'(x) cos(x) g(x) cos(x) rarr; g'(x) -sin(x)

Therefore:

Derivative of y sin(x)cos(x)

dy/dx cos(x)cos(x) sin(x)(-sin(x)) cos^2(x) - sin^2(x)

This expression can be further simplified using the double-angle identity for cosine:

cos(2x) cos^2(x) - sin^2(x)

Thus, the derivative is:

dy/dx cos(2x)

Using Trigonometric Identities

A more straightforward approach involves recognizing a trigonometric identity:

sin(2x) 2sin(x)cos(x)

From this identity, we can rewrite y sin(x)cos(x) as:

y 1/2 sin(2x)

To find the derivative, we apply the chain rule:

dy/dx 1/2 (2cos(2x)) cos(2x)

Deriving Using First Principles

First principles involve the definition of a derivative:

dy/dx lim{h→0} [(f(x h) - f(x)) / h]

For y sin(x)cos(x), this becomes:

dy/dx lim{h→0} [(sin(x h)cos(x h) - sin(x)cos(x)) / h]

Using the identity sin(x y) sin(x)cos(y) cos(x)sin(y), we expand sin(x h)cos(x h):

dy/dx lim{h→0} [(sin(x)cos(h)cos(x) cos(x)sin(h)cos(x) - sin(x)cos(x)) / h]

Simplifying:

dy/dx lim{h→0} [(sin(x)cos(h)cos(x) cos(x)sin(h)cos(x) - sin(x)cos(x)) / h]

This can be further simplified using the small angle approximation for sin(h) ≈ h as :

dy/dx lim{h→0} [cos(x)cos(x)]

Therefore, the derivative is:

dy/dx cos(2x)

Conclusion

The derivative of y sin(x)cos(x) is cos(2x). This can be derived using the product rule, trigonometric identities, or first principles. Understanding these methods enhances our ability to tackle more complex problems in calculus.

Keywords: derivative of sin(x)cos(x), product rule, chain rule, trigonometric identities