Discovering the Relationship Between Variables | Exploring Implicit Functions and Implicit Differentiation

Implicit functions

Implicit functions refer to mathematical equations that establish a relationship between variables without explicitly solving for one variable in terms of the others

Implicit functions refer to mathematical equations that establish a relationship between variables without explicitly solving for one variable in terms of the others. In other words, the dependent variable is not expressed as an explicit function of the independent variables. Instead, the equation combines the variables and their derivatives in a way that does not allow for a direct expression of one variable in terms of the others.

For example, consider the equation:
x^2 + y^2 – 1 = 0

This equation represents a circle centered at the origin with a radius of 1. It is an implicit function because we cannot directly solve for y in terms of x or vice versa. However, we can still extract information about the relationship between x and y using implicit differentiation.

To differentiate the equation with respect to x, we treat y as an implicit function of x and apply the chain rule. The derivative of y^2 is 2yy’, where y’ represents dy/dx. Similarly, the derivative of x^2 is 2x. The derivative of the constant 1 is 0. Thus, the derivative of the equation with respect to x becomes:
2x + 2yy’ = 0

Now, we can solve for y’ by isolating the term involving y’:
2yy’ = -2x
y’ = -x / y

Although we were not able to express y explicitly in terms of x, we obtained the derivative of y with respect to x. This demonstrates how implicit functions allow us to analyze the relationship between variables even when we cannot solve for one explicitly. Implicit functions are particularly useful in cases where explicit solutions are not easily attainable or when investigating relationships involving multiple variables.

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