What is the difference between implicit and explicit differentiation?

Implicit and explicit differentiation are two different techniques used in calculus to find derivatives of functions. They are employed in different situations and have distinct methods of application:

Explicit Differentiation:

Explicit differentiation is the most common and straightforward method for finding the derivative of a function, especially when the function is expressed explicitly in terms of a single variable, typically y = f(x).
In explicit differentiation, you directly express y as a function of x and then differentiate it with respect to x using standard differentiation rules, such as the power rule, product rule, quotient rule, etc.
For example, if you have y = x^2 + 3x + 2, you can explicitly differentiate it by applying the power rule to each term: dy/dx = 2x + 3.
Implicit Differentiation:

Implicit differentiation is used when a function is defined implicitly, meaning it's not expressed explicitly as y = f(x), but rather as an equation involving both x and y.
To find the derivative using implicit differentiation, you differentiate both sides of the equation with respect to x while treating y as an implicit function of x.
When you differentiate terms involving y, you use the chain rule to account for the fact that y is a function of x. In other words, you differentiate y with respect to x and multiply it by dy/dx.
For example, if you have the equation x^2 + y^2 = 1, you can use implicit differentiation to find dy/dx: 2x + 2yy' = 0, and then solve for dy/dx.
In summary, explicit differentiation is used when you have a function explicitly expressed as y = f(x), while implicit differentiation is employed when the relationship between x and y is defined implicitly as an equation involving both variables. Implicit differentiation requires the use of the chain rule to differentiate y with respect to x, making it a useful technique in cases where it's not easy to isolate y explicitly.