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Derivatives in Polar Coordinates Calculator

Polar tangent-slope calculator. Enter r(θ) and θ to compute dy/dx=(r′sinθ+r cosθ)/(r′cosθ−r sinθ)—for polar curves.

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dy/dx = (r′sinθ + r cosθ)/(r′cosθ − r sinθ)
Use t for θ, e.g. 1+cos(t)
e.g. π/4≈0.785

📖 Tutorial｜Derivatives in Polar Coordinates

1. Definition
For polar r=r(θ), with x=r cosθ, y=r sinθ, the slope is dy/dx=(r′sinθ+r cosθ)/(r′cosθ−r sinθ).
2. Symbols
SymbolMeaning
r(θ)Polar radius
r′Derivative w.r.t. θ
θAngle (radians)
dy/dxTangent slope
3. How it works
  • We numerically compute r and r′;
  • Plug into the polar derivative formula;
  • Get the Cartesian tangent slope.
4. Steps
  1. Enter r(θ) using t for θ;
  2. Enter θ in radians;
  3. Click Calculate.
5. Example
Example: r=1 (circle) at θ=π/4. r′=0, dy/dx=−cot(π/4)=−1.
6. Pitfalls
Use t for the angle variable inside r(θ);
When the denominator is 0 the tangent is vertical;
θ must be in radians.

❓ FAQ｜Derivatives in Polar Coordinates

How is the polar slope formula derived?
Differentiate x=r cosθ and y=r sinθ w.r.t. θ, then divide (dy/dθ)/(dx/dθ).
Why use t for θ inside r?
The parser uses the variable name t for the angle.
What does a zero denominator mean?
The tangent is vertical and the slope is infinite.
Slope of the circle r=1?
dy/dx=−cotθ, perpendicular to the radius; at θ=π/4 it is −1.
Can θ be in degrees?
No, always use radians or the trigonometry will be wrong.