Tangent Line Equation Generator
Powers ^ · multiply with * or a space · sqrt( ) ln( ) log( ) sin( ) cos( ) tan( ) e pi |x| · implicit products like 2x or x sin x work
Try an example
A number or a constant such as pi/4
Tangent line
y = 6x - 9Point-slope form:
Working
Normal line
Perpendicular to the tangent at the same point (slope −1/m).
Linear approximation
Near x = 3, f(x) ≈ f(a) + m(x − a). For x = 3.1:
- Tangent estimate: 9.6
- Actual f(3.1): 9.61
- Error: 0.01
Graph
Find the equation of the tangent line to y = f(x) at x = a, in slope-intercept and point-slope form, together with the normal line and the linear approximation the tangent gives near the point.
Formula
Worked Example
f(4) = 2 and f′(x) = 1/(2√x), so the slope at 4 is 1/4 and the tangent is y − 2 = ¼(x − 4), or y = x/4 + 1. Used as an approximation it gives √4.1 ≈ 2.025, against the true 2.02485.
Frequently Asked Questions
How do you find the equation of a tangent line?
Differentiate f, then find the point f(a) and the slope m = f′(a), and put them into point-slope form: y − f(a) = m(x − a). For f(x) = x² at x = 3: f(3) = 9 and m = 6, so y = 6x − 9.
What is the normal line?
The line through the same point perpendicular to the tangent, with slope −1/f′(a). Where the tangent is horizontal (f′(a) = 0), the normal is the vertical line x = a.
What is linear approximation used for?
Near x = a, f(x) ≈ f(a) + f′(a)(x − a), so a hard function can be estimated with a straight line. The error grows with the distance from a and with how sharply the curve bends.