An implicit function is a function in which the dependent variable is not written directly in terms of the independent variable. Instead, the variables are connected in a single equation.
In general, an implicit function is written in the form: f(x, y) = 0 , where both x and y appear in the same equation, and y cannot be easily expressed as y = f(x).
In contrast, a function written directly as y = f(x) is called an explicit function.
Example: The unit circle can be represented by the implicit equation: x² + y² − 1 = 0
If the values of x are limited to the interval [−1, 1] and y is restricted to either only positive or only negative values, then the equation defines y as an implicit function of x. In this case, y can be treated as the value of a single-variable function, written as y = f(x).
If the implicit equation is rearranged for y, the two explicit forms of the function are: y = \pm \sqrt{1 - x^2}
Derivative of Implicit Functions
Implicit differentiation is used to differentiate equations in which x and y are written together in a single equation.
How to Differentiate an Implicit Function
Since y cannot be isolated easily, both sides of the equation are differentiated with respect to x using the chain rule wherever y appears (since y depends on x). Then we rearrange the result to solve for dy/dx.
Steps:
Differentiate both sides of the equation f(x, y) = 0 with respect to x.
Apply the chain rule to terms containing y, since y is treated as a function of x. Therefore, their derivatives include dy/dx.
Move all terms containing dy/dx to one side of the equation.
Solve algebraically to find dy/dx
Example: Find dy/dx for x² + y² = 25
Differentiating both sides with respect to x:
2x + 2y(dy/dx) = 0
Solving for dy/dx:
dy/dx = −x/y
The result contains both x and y, which is normal for implicit differentiation.
Properties
Not written directly in the form y = f(x).
Usually represented as an equation involving two or more variables, such as f(x, y) = 0.
Dependent and independent variables remain combined in the same equation.
Commonly represent non-linear relationships between variables.
A single value of x may correspond to more than one value of y.
Useful for representing curves such as circles and ellipses.
Their derivatives are found using implicit differentiation and the chain rule.
Real-Life Applications of Implicit Functions
Implicit functions have many important applications in mathematics and various real-world fields:
Used in geometry to represent curves such as circles, ellipses, and other complex shapes.
In physics and engineering, they help describe relationships between variables that cannot be expressed directly.
In computer graphics, implicit equations are used to model curves and surfaces.
In economics, implicit functions are used to study relationships between quantities such as cost, demand, and profit.
Implicit differentiation is used to find rates of change and slopes in real-world mathematical models.
Solved Examples
Example 1: Find the derivative of the implicit function x² + y² + 6xy + 3 = 0 with respect to x.
Solution:
Differentiating both sides with respect to x:
d/dx(x²) + d/dx(y²) + d/dx(6xy) + d/dx(3) = 0
2x + 2y(dy/dx) + 6(x·dy/dx + y) + 0 = 0
2x + 2y·dy/dx + 6x·dy/dx + 6y = 0
2x + 6y + (2y + 6x)·dy/dx = 0
2(x + 3y) + 2(y + 3x)·dy/dx = 0
(x + 3y) + (3x + y)·dy/dx = 0
(3x + y)·dy/dx = −(x + 3y)
dy/dx = −(x + 3y) / (3x + y)
Example 2: Find the derivative of the implicit function x + cos(xy) − y = 0 with respect to x.
