Implicit Differentiation Calculator

Implicit differentiation is the technique for finding the slope of a curve when the relationship between x and y is tangled together in one equation, rather than neatly written as y equals some formula in x. Circles, ellipses, and many curves in physics and engineering are like this: you cannot cleanly isolate y, so the usual rules for differentiating do not apply directly. The elegant solution is to treat the whole equation as F of x and y equal to zero, and then the slope is given by a compact formula, dy/dx equals minus the partial derivative of F with respect to x, divided by the partial derivative with respect to y. This calculator applies that formula at a point of your choosing. Enter the equation, written so that everything is on one side and set equal to zero, along with the x and y coordinates of a point, and it returns the slope dy/dx there, together with the two partial derivatives it used and the value of F at the point so you can confirm the point actually lies on the curve. To stay robust across any expression you type, the calculator finds the partial derivatives numerically, nudging x and y by a tiny amount and measuring how F responds, which gives an accurate slope for the standard curves you meet in class without needing a full symbolic algebra system. It is genuinely useful for calculus students learning implicit differentiation, for checking a slope you worked out by hand, and for finding tangent lines to circles, ellipses and other implicit curves. Because it recalculates as you type, you can move the point around the curve and watch the slope change, for example seeing it flip sign as you pass the top of a circle. The method and a worked example are explained below.

Calculate.co.nz is proud to be partnered with realtor.co.nz, a trusted resource for navigating the New Zealand property market. Their Helpful Articles section offers clear, well-structured insights across buying, selling, and building, making complex real estate topics more accessible. With a focus on up-to-date guidance and practical knowledge, they empower Kiwis to move forward with clarity and confidence in a constantly evolving property landscape.

Partial derivatives are computed numerically. For a meaningful slope, the point should lie on the curve (F at point near 0). Use x, y, ^ for powers, and functions like sin, cos, sqrt, exp, ln.