What is the Derivative of cos(xy) with Respect to x?

In mathematical analysis, in particular, in multivariate calculus, finding the derivative of functions involves understanding partial derivatives and the chain rule. Let's explore the derivative of the function (cos(xy)) with respect to (x).

Applying the Chain Rule to cos(xy)

To find the derivative of (cos(xy)) with respect to (x), we use the chain rule. Suppose (y) is a function of (x), the derivative is given by:

[ frac{d}{dx} cos(xy) -sin(xy) cdot frac{d}{dx}(xy) ]

Next, we differentiate (xy) with respect to (x) using the product rule:

[ frac{d}{dx}(xy) 1 frac{dy}{dx} ]

Substituting this back into our derivative expression, we obtain:

[ frac{d}{dx} cos(xy) -sin(xy) cdot left(1 frac{dy}{dx}right) ]

Further Considerations

When (y) is a function of (x), the derivative is:

[ frac{d}{dx} cos(xy) -sin(xy) left(1 frac{dy}{dx}right) ]

Partial Derivative

Often, we encounter functions of several variables and need to find partial derivatives. For the function (f(x, y) cos(xy)):

[ frac{delta f}{delta x} -sin(xy)frac{dy}{dx} ] [ frac{delta f}{delta y} -xsin(xy)left(1 frac{dy}{dx}right) ]

If we know the functional relationship between (y) and (x), we can find (frac{dy}{dx}), and the total derivative with respect to (x) becomes:

[ frac{df}{dx} frac{delta f}{delta x} cdot frac{dy}{dx} frac{delta f}{delta y} cdot left(1 frac{dy}{dx}right) ]

Simplifying further:

[ frac{df}{dx} -sin(xy) left(1 frac{dy}{dx}right) ]

Inverse Trigonometric Method

For an alternative method, one can use the inverse trigonometric functions. The cosine function and its inverse sine function are related, and we can leverage this relationship to further analyze the derivative.

For the function (f(x, y) cos(xy)), the derivative with respect to (x) remains the same whether we use the chain rule or inverse trigonometric identities:

[ frac{df}{dx} -sin(xy) left(1 frac{dy}{dx}right) ]