Tangent Line Calculator
Find the tangent line to a curve at any point, in slope-intercept form.
Enter your values
Enter the f(a) — value of the function at a you are working with.
Enter the f′(a) — derivative at a you are working with.
Enter the a — the x value you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- 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).
y = f(a) + f′(a)(x − a)
At a glance
| What it does | Find the tangent line to a curve at any point, in slope-intercept form. |
|---|---|
| Category | Math |
| Inputs needed | f(a) — value of the function at a, f′(a) — derivative at a, a — the x value |
| Main output | Tangent line |
| Formula | y = f(a) + f′(a)(x − a) |
| Cost | Free — no sign-up, no download |
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.
Inputs explained
- f(a) — value of the function at a
- Enter the f(a) — value of the function at a you are working with.
- f′(a) — derivative at a
- Enter the f′(a) — derivative at a you are working with.
- a — the x value
- Enter the a — the x value you are working with.
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
Results
- Tangent liney = 6x − 9
- Slope6
- y-intercept-9
- Point of tangency(3, 9)
- Normal line slope-0.1667
- Point-slope: y − 9 = 6(x − 3)
- Expand: y = 6x + -9
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).
What do I need to enter into the Tangent Line Calculator?
Just 3 values: f(a) — value of the function at a, f′(a) — derivative at a, a — the x value. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Tangent Line Calculator work out the 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). It applies the formula y = f(a) + f′(a)(x − a) and shows the working so you can check each step by hand.
Is the Tangent Line Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.