What is an implicit derivative?
Implicit differentiation is used when an equation defines y indirectly in terms of x, rather than giving y as an explicit function such as y = f(x). Instead of solving the equation for y first, differentiate the relationship with respect to x and then solve for dy/dx.
The implicit differentiation formula
If an equation can be written as F(x,y) = 0, then
dy/dx = −Fx / Fy
provided Fy is not zero at the point.
How to use the implicit derivative calculator
Enter the complete equation in x and y, then enter the x and y coordinates where you want the derivative. The calculator converts the equation into F(x,y) = 0, differentiates it with respect to x and y, and evaluates the resulting formula at your point.
Example: x2 + y2 = 25 at (3, 4)
Fx = 2x = 6 and Fy = 2y = 8.
Therefore dy/dx = −6/8 = −0.75.
Supported expressions
You can use numbers, x, y, parentheses, +, −, *, / and ^. The calculator also supports common functions including sin, cos, tan, sqrt, log, exp and abs. Write multiplication explicitly, for example 2*x*y rather than 2xy.
Frequently asked questions
What is an implicit differentiation calculator?
It is an online tool that calculates dy/dx for an equation involving both x and y. This is useful when y is not isolated on one side of the equation.
What does dy/dx mean?
dy/dx is the derivative of y with respect to x. At a particular point on a curve, it represents the slope of the tangent line when the derivative exists.
Why do I need both x and y values?
An implicit equation can have different derivative values at different points. Supplying both coordinates lets the calculator evaluate the derivative at the point you care about.
What if Fy is zero?
The formula −Fx/Fy cannot be evaluated when Fy is zero. The curve may have a vertical tangent or require a different local analysis.