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Home›Function Tools›Parametric Derivative Calculator

Parametric Derivative Calculator

Compute dy/dx and d^2y/dx^2 for parametric curves x(t), y(t)

Select Curve Type

Linear-Linear (Line) Linear-Quadratic (Parabola)

x(t)=at+b, y(t)=ct+d

x(t)=t+, y(t)=t+

Parametric Derivative Formulas

dy/dx = (dy/dt) / (dx/dt)

d^2y/dx^2 = d/dt(dy/dx) / (dx/dt)

First: find x(t) and y(t)

Then: divide, differentiate, divide again

Parametric derivatives allow finding slopes of parametric curves without eliminating the parameter. The first derivative dy/dx gives the tangent slope. The second derivative gives concavity of the parametric curve.

⚠ The formula dy/dx = y/x requires dx/dt != 0. At points where dx/dt=0, the tangent is vertical.

What Are Parametric Derivatives?

Parametric equations describe curves using a parameter t. The derivative dy/dx gives the slope of the tangent to the parametric curve. The chain rule directly gives dy/dx = (dy/dt)/(dx/dt). The second derivative reveals concavity.

First Derivative

dy/dx = y/x. Differentiate both parametric equations, then divide y by x. Gives slope at any t.

Second Derivative

d^2y/dx^2 = (d/dt(y/x))/x. Differentiate the first derivative with respect to t, then divide by x again.

Tangent Lines

At parameter t0, slope = dy/dx at t0. Tangent line: y-y(t0)=m*(x-x(t0)). Horizontal when y=0, vertical when x=0.

Applications

Motion analysis (velocity vector = (x,y)), curve sketching, arc length, surface area of revolution.

Teaching Example: x(t)=2t+1, y(t)=3t-2. x=2, y=3. dy/dx=3/2=1.5. d^2y/dx^2 = d/dt(1.5)/2 = 0/2 = 0. The curve is a line (constant slope, no concavity).

Applications

Calculus Motion Physics Computer Graphics Robotics

Frequently Asked Questions

Parametric dy/dx formula?▼

dy/dx = (dy/dt)/(dx/dt). Differentiate x(t) and y(t) separately, then divide y by x.

Second derivative parametric?▼

d^2y/dx^2 = d/dt(dy/dx) / dx/dt. Differentiate dy/dx w.r.t t, then divide by x.

Horizontal tangent?▼

Horizontal when dy/dt=0 (and dx/dt != 0). Vertical when dx/dt=0 (and dy/dt != 0).

Parametric vs explicit?▼

Parametric curves can represent functions that fail the vertical line test (circles, loops). Parametric derivatives handle all cases.

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