Tangent Line Calculator
Find the tangent line to a curve at any point, in slope-intercept form.
Reviewed by James Whitfield · How we check our maths
Your values
ⓘ Live results
Results
- Tangent line
- y = 6x − 9
- Slope
- 6
- y-intercept
- -9
- Point of tangency
- (3, 9)
- Normal line slope
- -0.1667
Working
- 1Point-slope: y − 9 = 6(x − 3)
- 2Expand: y = 6x + -9
Quick answer
The tangent line at x = a has slope f′(a) and passes through (a, f(a)), so its equation is y = f(a) + f′(a)(x − a).
The formula
y = f(a) + f′(a)(x − a)
Worked example
Worked example
Using the values the calculator loads with.
Inputs
- f(a) — value of the function at a9
- f′(a) — derivative at a6
- a — the x value3
Result
- Tangent liney = 6x − 9
- Slope6
- y-intercept-9
- Point of tangency(3, 9)
- Normal line slope-0.1667
How to use the Tangent Line Calculator
- 1Differentiate the function and evaluate it at your chosen x to get f′(a).
- 2Evaluate the original function at the same x to get f(a).
- 3Enter both plus the x value to read the finished equation.
Frequently asked questions
What is the tangent line used for?
It is the best straight-line approximation of a curve near a point — the basis of linearisation, Newton's method and marginal rates in economics.
What is the normal line?
The line perpendicular to the tangent at the same point; its slope is −1 ÷ f′(a).
About this calculator
- Built and reviewed by James Whitfield · How we build and test calculators
- Report an error