Implicit Differentiation Calculator
Finds the slope dy/dx of a conic section curve (circle or ellipse) defined by Ax² + By² = C at a specified point. Use it when the relationship between x and y cannot be solved explicitly for y.
About this calculator
Implicit differentiation finds dy/dx for a curve defined by an equation F(x, y) = 0, without solving explicitly for y. For a general conic Ax² + By² = C, differentiating both sides with respect to x gives 2Ax + 2By·(dy/dx) = 0. Solving for dy/dx yields: dy/dx = −(Ax) / (By). This formula applies to circles (A = B) and ellipses (A ≠ B) alike. The slope is undefined when y = 0, corresponding to horizontal tangent points where the curve has vertical tangents. Implicit differentiation is a consequence of the chain rule: whenever y appears in an expression differentiated with respect to x, a factor dy/dx is attached. This technique extends to far more complex implicit relationships in physics and engineering.
How to use
Consider the ellipse 4x² + 9y² = 36. Set xCoeff = 4, yCoeff = 9, constant = 36. Find the slope at the point (1, √(32/9)) ≈ (1, 1.886). Using dy/dx = −(Ax)/(By) = −(4 · 1)/(9 · 1.886) = −4/16.97 ≈ −0.236. This means the ellipse has a downward slope of about −0.236 at that point. You can verify by checking that (1, 1.886) satisfies the original equation: 4(1) + 9(3.556) ≈ 4 + 32 = 36. ✓ Enter xPoint = 1 and yPoint = 1.886 to get this result directly.
Frequently asked questions
What is implicit differentiation and why is it needed for curves like circles?
Implicit differentiation is a technique for finding dy/dx when y is not expressed explicitly as a function of x — for example, in x² + y² = 25. Solving for y gives ±√(25 − x²), two separate branches, each valid only for part of the curve. Implicit differentiation avoids this split by differentiating the equation as written, applying the chain rule to y terms. The result dy/dx = −x/y works for the entire circle except where y = 0. This makes it essential for analyzing circles, ellipses, hyperbolas, and more complex algebraic curves.
When is the slope dy/dx undefined for an implicitly defined curve?
The slope dy/dx = −(Ax)/(By) is undefined whenever the denominator By equals zero, which occurs at points where y = 0. Geometrically, these are points where the tangent line to the curve is vertical — the curve is momentarily moving straight up or down. For a circle x² + y² = r², the vertical tangents occur at (r, 0) and (−r, 0), the leftmost and rightmost points. At these locations dy/dx does not exist, meaning the traditional slope concept breaks down. You can instead compute dx/dy at such points, which equals zero (a well-defined horizontal rate of change).
How does implicit differentiation relate to the chain rule in calculus?
Implicit differentiation is a direct application of the chain rule. When you differentiate an equation involving y with respect to x, every occurrence of y must be treated as a composite function y(x). By the chain rule, d/dx[f(y)] = f'(y)·(dy/dx). For example, d/dx[y²] = 2y·(dy/dx). Collecting all dy/dx terms on one side and solving algebraically gives the final derivative formula. This principle extends beyond conics — it applies to any implicitly defined relation, including trigonometric, exponential, and logarithmic equations mixed with polynomial terms.