Equation of Tangent Line Calculator
Find the equation of a curve’s tangent line at a point using function input, point value, and derivative for accuracy.
Enter Values
Equation Preview
Helping Notes
Enter only the shown fields; others are hidden because they’re unnecessary for the selected form.
Supported functions include sin, cos, tan, exp, ln, sqrt, and constants like pi and e.
Polar uses x = r cos t, y = r sin t. Implicit slope uses −Fx/Fy when F(x,y)=0.
Results
Point on Curve
Tangent Slope (m)
Tangent Line (point-slope)
Tangent Line (slope-intercept)
Quick Check
What Is a Tangent Line Calculator?
A Tangent Line Calculator constructs the best linear approximation to a curve at a chosen point. For a differentiable function at , the slope is and the tangent line is . This line matches the curve’s value and instantaneous rate of change at , enabling fast estimates near that point via the linearization . The calculator accepts explicit, parametric, and implicit inputs, returns symbolic expressions when possible, numeric evaluations otherwise, and presents steps clearly so you can verify differentiation and substitution.
About the Tangent Line Calculator
The engine differentiates using exact rules (power, product, quotient, chain) or numeric differences if necessary. For parametric curves , it computes (when ) and evaluates at . For implicit curves , it uses at the point, provided . Slope‑intercept and point‑slope forms are both displayed, and the linearization is highlighted for approximation tasks, error bounds, and Newton’s method initialization.
How to Use This Tangent Line Calculator
- Enter the function (explicit , parametric, or implicit) and specify the point or parameter value.
- Compute derivatives automatically. The tool finds , builds , and simplifies to .
- Optionally show linearization and use it to estimate nearby values.
- Copy the final equation and steps for assignments, checks, or further numerical methods.
Examples
- Explicit: at . Slope . Tangent: .
- Trig: at . Slope . Tangent: .
- Parametric: at . Slope . Point . Tangent: .
- Implicit: at . Slope . Tangent: .
Formula Snippets Ready for Rendering
FAQs
What is the tangent line used for?
It gives the best linear approximation near a point and underlies linearization, Newton’s method, and instantaneous rate calculations.
How do I find the slope quickly?
Differentiate the function and evaluate at the point: . For parametrics, use .
What if the derivative does not exist?
At cusps, corners, or vertical tangents, no unique tangent line exists; the calculator will flag non‑differentiability.
Can I input an implicit curve?
Yes—provide and a point; slope is when .
Does the tool show both point‑slope and slope‑intercept forms?
It presents and simplifies to for easy graphing.
How accurate are numeric derivatives?
Finite‑difference estimates are close for smooth functions; the tool adapts step size to control truncation and rounding error.