25
0
0
1
-0.75
-0.75
6.25
1.333333
0
-0.390625
0.2
4
25
0
0
1
-0.75
-0.75
6.25
1.333333
0
-0.390625
0.2
4
The Implicit Derivative Calculator computes $$\frac{dy}{dx}$$ for the implicit equation $$ax^2 + by^2 = c$$ at any specified point $$(x, y)$$. Implicit differentiation is essential when a relationship between x and y cannot be easily solved for y as an explicit function of x. This calculator finds the derivative, tangent line, normal line, second implicit derivative, and curvature at the given point, while also verifying that the point lies on the curve.
Most curves studied in introductory calculus are given explicitly as $$y = f(x)$$. But many important curves are defined implicitly by equations like $$x^2 + y^2 = 25$$ (a circle), $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ (an ellipse), or more complex relationships. For these curves, it may be impossible or inconvenient to solve for y in terms of x — the equation may define y as a multi-valued function of x (like the upper and lower halves of a circle).
Implicit differentiation solves this problem by differentiating both sides of the equation with respect to x, using the chain rule to handle y-terms. Whenever we differentiate a term containing y, we multiply by $$\frac{dy}{dx}$$ (since y is implicitly a function of x). For the equation $$ax^2 + by^2 = c$$:
$$2ax + 2by \cdot \frac{dy}{dx} = 0$$
Solving for $$\frac{dy}{dx}$$:
$$\frac{dy}{dx} = -\frac{ax}{by}$$
This formula is valid at any point on the curve where $$by \neq 0$$. At points where $$y = 0$$ (the x-axis intersections), the tangent line is vertical and the derivative is undefined.
The equation $$ax^2 + by^2 = c$$ encompasses several important curves depending on the signs of a and b. When $$a > 0$$ and $$b > 0$$, it defines an ellipse (or circle if $$a = b$$). When one is positive and the other negative, it defines a hyperbola. These conic sections appear throughout physics, engineering, and astronomy — planetary orbits are ellipses, hyperbolas describe the paths of unbound objects in gravitational fields, and circles are fundamental to rotational mechanics.
Beyond the first derivative, this calculator computes the second implicit derivative $$\frac{d^2y}{dx^2}$$, which measures the concavity of the implicit curve. It also computes the curvature $$\kappa$$, which quantifies how sharply the curve bends. Curvature is essential in road design (banking of curves), optics (lens and mirror shapes), and differential geometry.
Enter the equation coefficients and a point (x, y), and the calculator will verify whether the point lies on the curve, then compute the implicit derivative and all related geometric quantities.
The calculator performs implicit differentiation on the equation $$ax^2 + by^2 = c$$.
Step 1: Verify the point. Check whether $$ax^2 + by^2 = c$$ holds at the given $$(x, y)$$.
Step 2: Implicit differentiation. Differentiate both sides with respect to x:
$$\frac{d}{dx}[ax^2 + by^2] = \frac{d}{dx}[c]$$
$$2ax + 2by \cdot \frac{dy}{dx} = 0$$
$$\frac{dy}{dx} = -\frac{ax}{by}$$
Step 3: Tangent line. At $$(x_0, y_0)$$: $$y = \frac{dy}{dx} \cdot x + B$$ where $$B = y_0 - \frac{dy}{dx} \cdot x_0$$.
Step 4: Normal line. The normal is perpendicular to the tangent: slope = $$-1 / (dy/dx) = \frac{by}{ax}$$.
Step 5: Second implicit derivative. Differentiate $$\frac{dy}{dx} = -\frac{ax}{by}$$ again using the quotient rule:
$$\frac{d^2y}{dx^2} = -\frac{a(by \cdot 1 - x \cdot b \cdot \frac{dy}{dx})}{(by)^2} = -\frac{a(by^2 + ax^2)}{b^2 y^3}$$
where we substituted $$\frac{dy}{dx} = -\frac{ax}{by}$$.
Step 6: Curvature.
$$\kappa = \frac{|d^2y/dx^2|}{(1 + (dy/dx)^2)^{3/2}}$$
Point on Curve? verifies whether the given (x,y) satisfies the equation ax²+by²=c. If the point is not on the curve, the derivative still computes at that (x,y) but represents the derivative of a different level curve passing through that point.
dy/dx is the slope of the curve at the given point. It is the implicit derivative obtained by treating y as a function of x. A positive value means y increases as x increases along the curve; negative means y decreases.
Tangent Line (slope and y-intercept) defines the line that just touches the curve at the given point. The tangent line is the best linear approximation of the curve near that point.
Normal Line is perpendicular to the tangent line. In physics, the normal direction is important for computing forces on curved surfaces, reflection of light, and the centripetal direction in circular motion.
d²y/dx² describes how the slope of the curve changes — the concavity of the implicit curve. Combined with dy/dx, it completely characterizes the local shape of the curve.
Curvature $$\kappa$$ measures how sharply the curve bends. Larger values mean tighter bending. For a circle of radius R, $$\kappa = 1/R$$ everywhere. The radius of curvature is $$1/\kappa$$.
Inputs
Results
3² + 4² = 25, confirming the point is on the circle. dy/dx = −3/4 = −0.75. The normal line passes through the origin (as expected for a circle centered at the origin). Curvature = 0.2 = 1/5, consistent with radius 5.
Inputs
Results
At the top of the ellipse (0,2), the tangent is horizontal (slope 0), so the normal is vertical. dy/dx = −4(0)/(9(2)) = 0. d²y/dx² = −4(36)/(729·8) = −4/18 ≈ −0.222, confirming the curve is concave down at its peak.
Implicit differentiation is a technique for finding dy/dx when y is defined implicitly by an equation F(x,y) = 0 rather than explicitly as y = f(x). You differentiate both sides of the equation with respect to x, applying the chain rule to every y-term (multiplying by dy/dx), then solve algebraically for dy/dx. It works for any differentiable implicit relationship.
When y = 0, the formula dy/dx = −ax/(by) involves division by zero. Geometrically, this corresponds to a point where the tangent line is vertical — the curve is going straight up or down, so the slope is infinite. At such points, dx/dy would be well-defined (and equal to zero) instead.
When a and b are both positive and c > 0: an ellipse (or circle if a = b). When a and b have opposite signs and c ≠ 0: a hyperbola. When c = 0: the equation reduces to a point or pair of lines. The coefficients a and b control the shape and orientation of the conic section.
The normal line is perpendicular to the curve at the point of tangency. In physics, it gives the direction of the centripetal force for circular motion and the direction of reflection for light or sound off a curved surface. In differential geometry, the normal direction is fundamental to computing curvature and defining coordinate systems on curves.
Curvature and radius of curvature are reciprocals: $$\kappa = 1/R$$. A circle of radius 5 has curvature 0.2 everywhere. For a general curve, the curvature varies from point to point. The osculating circle at a point is the circle that best approximates the curve there; its radius is the radius of curvature 1/κ.
Strictly speaking, the implicit derivative at a point only has geometric meaning if the point lies on the curve ax² + by² = c. If the point is off the curve, the computed derivative corresponds to the level curve ax² + by² = k that passes through (x,y), where k = ax² + by² ≠ c. The calculator warns you when the point is not on the specified curve.
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