Implicit differentiation is used when an equation contains x and y together and y has not been isolated as an explicit function of x. The Implicit Differentiation Calculator helps carry out the differentiation while treating y as a quantity that depends on x.
How Implicit Differentiation Works
Both sides of the equation are differentiated with respect to x. Whenever a term contains y, its derivative must account for dy/dx. For example, differentiating yΒ² gives 2y(dy/dx), not simply 2y.
Using the Calculator
Enter the equation in the form supported by the tool and review the requested variables before calculating. Make sure multiplication, powers, and parentheses are entered clearly. The returned derivative can then be rearranged to isolate dy/dx when the expression is not already in that form.
Example
Take xΒ² + yΒ² = 25. Differentiating both sides gives 2x + 2y(dy/dx) = 0. Solving for dy/dx produces βx/y. This result gives the slope of the curve at points where y is nonzero.
Where It Is Useful
Implicit differentiation is common with circles, related curves, and equations where solving explicitly for y would introduce unnecessary square roots or multiple branches. It also provides a direct way to study the slope of a curve at a particular point.
Check the Output
After getting a result, substitute a known point into the original equation when possible. This confirms that the point belongs to the curve before you interpret the derivative there.