Understanding the Derivative of xcosx / (x - sinx)

When dealing with calculus, particularly with differentiation, understanding the techniques and rules involved is crucial. In this article, we delve into finding the derivative of the expression ycosx / (x - sinx). This expression involves a quotient of trigonometric functions, and mastering the quotient rule of differentiation is key.

The Quotient Rule of Differentiation

The quotient rule of differentiation is a fundamental concept that allows us to find the derivative of a function that is in the form of a fraction. If we have a function y f(x) / g(x), then its derivative (dy/dx) can be found using the formula:

[ frac{dy}{dx} frac{g(x) cdot f'(x) - f(x) cdot g'(x)}{[g(x)]^2} ]

In the case of the expression ycosx / (x - sinx), we will need to apply this rule accurately. Let's break it down step by step.

Applying the Quotient Rule

Given the function y xcosx / (x - sinx), let's denote it as:

[ f(x) x cos x ] [ g(x) x - sin x ]

We need to find the derivatives of f(x) and g(x). Let's start with f(x) first:

[ f'(x) frac{d}{dx}(x cos x) ] [ x cdot (-sin x) cos x ] [ -x sin x cos x ]

Now, let's find the derivative of g(x):

[ g'(x) frac{d}{dx}(x - sin x) ] [ 1 - cos x ]

Now that we have both derivatives, we can apply the quotient rule:

[ frac{dy}{dx} frac{(x - sin x) cdot (-x sin x cos x) - (x cos x) cdot (1 - cos x)}{(x - sin x)^2} ]

Expanding this expression:

[ frac{dy}{dx} frac{(-x^2 sin x x cos x - x sin x sin x - sin x cos x) - (x cos x - x cos^2 x)}{(x - sin x)^2} ]

Simplifying the numerator:

[ frac{-x^2 sin x x cos x - x sin^2 x - sin x cos x - x cos x x cos^2 x}{(x - sin x)^2} ]

Factoring out common terms:

[ frac{-x^2 sin x - x sin^2 x x cos^2 x}{(x - sin x)^2} ]

We can rewrite sin^2 x cos^2 x as 1:

[ frac{-x^2 sin x - x sin^2 x x (1 - sin^2 x)}{(x - sin x)^2} ]

Simplifying further:

[ frac{-x^2 sin x - x sin^2 x x - x sin^2 x}{(x - sin x)^2} ]

Combining like terms:

[ frac{-x^2 sin x - 2x sin^2 x x}{(x - sin x)^2} ]

[ frac{x(1 - x sin x - 2 sin^2 x)}{(x - sin x)^2} ]

Therefore, the final derivative is:

[ frac{dy}{dx} frac{x(1 - x sin x - 2 sin^2 x)}{(x - sin x)^2} ]

Conclusion

Understanding the derivative of complex expressions like x cos x / (x - sin x) can be challenging, but applying the quotient rule effectively can simplify the process. Mastery of differentiation techniques, including the quotient rule and handling trigonometric functions, is essential in advanced calculus and mathematics.

Related Keywords

derivative, quotient rule, trigonometric functions