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An explicit function is a function in which one variable (usually y) is expressed as a function of another variable (usually x). It is written in the form y = f(x). For example, y = 2x + 3 is an explicit function. An implicit function, on the other hand, is a function in which the relationship between x and y is not explicitly solved for y. It is written in a general form, such as F(x,y) = 0. For example, x^2 + y^2 = 1 implicitly defines a function because the relationship between x and y is not written as y = f(x).

To find the derivative of an explicit function, we use the basic rules of differentiation such as the power rule, product rule, quotient rule, and chain rule as needed. For example, given the explicit function y = 2x + 3, dy/dx = 2 because we can directly apply the power rule.

To compute the derivative of an implicit function, we use implicit differentiation. Here, we differentiate both sides of the equation with respect to x while treating y as a function of x. To account for this, whenever we differentiate a term containing y, we multiply by the derivative, dy/dx. For example, given the implicit function x^2 + y^2 = 1, we differentiate both sides with respect to x, yielding 2x + 2y(dy/dx) = 0. From this equation, we can solve for dy/dx.

The main differences between computing the derivatives of explicit and implicit functions are: 1. Explicit functions are written as y = f(x), while implicit functions are typically written in the form F(x,y) = 0. 2. When computing the derivative of an explicit function, we directly apply the appropriate differentiation rules, whereas, for implicit functions, we rely on implicit differentiation that involves differentiating both sides of the equation and multiplying by dy/dx when differentiating terms involving y. 3. Finding the derivative of an implicit function often requires additional steps to solve for dy/dx, unlike the direct approach used with explicit functions. 4. Implicit functions might describe relations where it would be difficult or awkward to represent the relationship between x and y explicitly (like an ellipse, for example).