Derivative of xy

We now understand the derivative of xy. It is commonly represented as d/dx (xy) or (xy)', and its value is y + x(dy/dx). The function xy has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Product Function: (xy = x · y). Product Rule: Rule for differentiating xy (since it consists of the product of two functions).

The derivative of xy can be denoted as d/dx (xy) or (xy)'. The formula we use to differentiate xy is: d/dx (xy) = y + x(dy/dx) The formula applies to all x where both x and y are differentiable functions of x.

We can derive the derivative of xy using proofs. To show this, we will use the rules of differentiation. There are several methods we use to prove this, such as: Using Product Rule We will now demonstrate that the differentiation of xy results in y + x(dy/dx) using the above-mentioned method: Using Product Rule To prove the differentiation of xy using the product rule, We use the formula: d/dx [u · v] = u' · v + u · v' Consider u = x and v = y So we get, xy = u · v By product rule: d/dx (xy) = (dx/dx) · y + x · (dy/dx) d/dx (xy) = y + x(dy/dx) Hence, proved.

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky. To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like xy. For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′(x) Similarly, the third derivative, f′′′(x) is the result of the second derivative, and this pattern continues. For the nth Derivative of xy, we generally use fⁿ(x) for the nth derivative of a function f(x), which tells us the change in the rate of change (continuing for higher-order derivatives).

When y is a constant, the derivative simplifies to x(dy/dx). When x is a constant, the derivative simplifies to y.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Product Rule: A differentiation rule used to differentiate functions that are the product of two other functions. First Derivative: The initial result of a function, which gives us the rate of change of a specific function. Higher-Order Derivative: Derivatives of a function taken multiple times, providing insights into the changing rates of change. Constant: A value that does not change and has no derivative with respect to a variable.