Derivative of 3xy

The derivative of the function 3xy with respect to x is found using the product rule of differentiation. The function 3xy represents a product of 3x and y, and its derivative can be denoted as d/dx (3xy). The key concepts to understand here include: - Product Rule: Used to differentiate products of two functions. - Constant Multiplication Rule: The derivative of a constant times a function is the constant times the derivative of the function.

To differentiate 3xy with respect to x, we apply the product rule. If y is a function of x, the derivative of 3xy is: d/dx (3xy) = 3(dy/dx)x + 3y This formula applies to all x where y is differentiable with respect to x.

We can derive the derivative of 3xy using the product rule. To demonstrate this, the following methods are used: - By Applying the Product Rule - Using Constant Multiplication Rule By Applying the Product Rule: The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Consider f(x) = 3x and g(x) = y. Thus, d/dx (3xy) = 3y + 3x(dy/dx) Using Constant Multiplication Rule: The constant multiplication rule states that a constant factor can be taken out of the differentiation operation. Thus, d/dx (3xy) = 3(d/dx (xy)) Applying the product rule on xy, we get: d/dx (xy) = y + x(dy/dx) Therefore, d/dx (3xy) = 3(y + x(dy/dx))

Higher-order derivatives involve differentiating a function multiple times. In the context of 3xy, the first derivative provides the rate of change, while the second derivative gives the rate of change of the rate of change, and so on. For the first derivative, we write f′(x), which indicates how the function changes. The second derivative, f″(x), is derived from the first derivative and gives insight into the curvature or concavity of the function. Similarly, the third derivative, f‴(x), is the derivative of the second derivative, and this pattern continues.

In some cases, specific values of x may lead to undefined derivatives or special conditions. - If y is constant, the derivative simplifies significantly. - If x or y is zero, the derivative may simplify to zero or another specific value.

Derivative: A measure of how a function changes as its input changes. Product Rule: A rule used to differentiate products of two functions. Constant Multiplication Rule: The derivative of a constant times a function is the constant times the derivative of the function. Higher-Order Derivatives: Derivatives of derivatives, giving additional insights into the behavior of functions. Chain Rule: A rule for differentiating compositions of functions.