Derivative of cos(xy)

The derivative of cos(xy) is expressed as d/dx (cos(xy)) or (cos(xy))'. Since cos(xy) is a product of two variables within the cosine function, we apply the chain rule for differentiation. The key concepts include: - Cosine Function: cos(xy) is a trigonometric function. - Chain Rule: A rule for differentiating composite functions. - Product Rule: A rule for differentiating products of two functions.

The derivative of cos(xy) with respect to x can be denoted as d/dx (cos(xy)) or (cos(xy))'. By applying the chain rule, we get: d/dx (cos(xy)) = -sin(xy) * (y + x * dy/dx) This formula applies to all x and y where xy is defined in the domain of the cosine function.

We can derive the derivative of cos(xy) using several methods. These include: - By First Principle - Using Chain Rule - Using Product Rule We will demonstrate the differentiation of cos(xy) using the following methods: Using Chain Rule To prove the differentiation of cos(xy) using the chain rule, we consider the composite function inside the cosine: f(x, y) = xy, so the differentiation involves: d/dx (cos(f(x, y))) = -sin(f(x, y)) * d/dx(f(x, y)) d/dx (cos(xy)) = -sin(xy) * (y + x * dy/dx) The derivative of f(x, y) with respect to x is y, and we apply the chain rule to get the final result. Using Product Rule To differentiate cos(xy) using the product rule, consider: g(x) = x and h(y) = y, hence f(x, y) = xy. d/dx (cos(xy)) = -sin(xy) * d/dx(xy) = -sin(xy) * (y + x * dy/dx) By applying the product rule, we determine the derivative of xy, then apply it in the chain rule to find the derivative of cos(xy).

When a function is differentiated multiple times, the resulting derivatives are referred to as higher-order derivatives. Higher-order derivatives are often more complex. To illustrate, consider a vehicle's acceleration (second derivative) and jerk (third derivative). Understanding higher-order derivatives of functions like cos(xy) can be challenging but is essential in fields like physics and engineering. For the first derivative, we write f′(x), indicating how the function changes at a point. The second derivative, f′′(x), is derived from the first, and this pattern continues for higher-order derivatives.

When x or y is such that xy equals π/2, the derivative is undefined because sin(xy) has a vertical asymptote there. When x or y is 0, the derivative of cos(xy) equals zero, as sin(0) is zero.

Derivative: A measure of how a function changes with respect to changes in its input variables. Cosine Function: A trigonometric function representing the cosine of an angle or product of variables. Chain Rule: A fundamental rule for differentiating composite functions. Product Rule: A rule for differentiating products of two functions. Partial Derivative: The derivative of a function with respect to one variable while keeping others constant.