derivative implicit calculator - Aaron Graves, PhDude Replica

Implicit equation (use x and y)

Tip: Use ^ for powers, sqrt(...), sin(...), cos(...), exp(...), and include an equals sign when possible.

Optional x-value for slope at a point

Optional y-value for slope at a point

Quick examples:

Enter an equation and click “Calculate dy/dx”.

What This Implicit Derivative Calculator Does

This tool computes dy/dx when y is defined implicitly by an equation involving both x and y, such as x² + y² = 25 or x³ + y³ = 6xy. Instead of solving explicitly for y first, the calculator uses the standard implicit differentiation identity:

If F(x, y) = 0, then dy/dx = -Fx / Fy

Here, Fx is the partial derivative of F with respect to x, and Fy is the partial derivative with respect to y.

How to Use It

1) Enter an equation

Write your relation in x and y. You can enter either:

A full equality, like x^2 + y^2 = 25, or
A single expression interpreted as expression = 0, like x^2 + y^2 - 25.

2) (Optional) Enter a point

If you provide both x and y values, the calculator evaluates the slope numerically at that point. This is useful for tangent-line problems.

3) Read the result steps

The output shows:

The transformed function F(x, y)
Fx and Fy
The symbolic formula for dy/dx
Optional numeric slope at the entered point

Why Implicit Differentiation Matters

Many important curves are difficult (or impossible) to write as a single explicit function y = f(x). Circles, ellipses, and many engineering constraints are naturally implicit. With implicit differentiation, you can still find slopes, tangent lines, and critical behavior directly from the relation itself.

Examples You Can Try

Example A: Circle

x^2 + y^2 = 25

The derivative is dy/dx = -x/y. At (3, 4), the slope is -0.75.

Example B: Nonlinear mixed equation

x^3 + y^3 = 6xy

This produces a richer derivative expression where both x and y remain present. Perfect for studying local behavior near special points.

Common Input Tips

Use * for multiplication when needed (for clarity, write 6*x*y).
Use parentheses in complex expressions, e.g., (x+y)^2.
Supported functions include sin, cos, tan, sqrt, log, and exp.
If Fy = 0 at a point, the slope may be undefined or vertical.

Final Thought

Implicit differentiation is one of the most practical ideas in calculus. Use this calculator to check homework, verify symbolic work, or explore how hidden relationships between x and y determine slope behavior.