How to find a tangent line
You need two things: a point and a slope. The point is (a, f(a)). The slope is the derivative evaluated at a, f′(a). Put them into the point-slope form of a line.
The formula
y − f(a) = f′(a)(x − a)Point-slope form, then solve for y to get y = mx + b.
Worked example
Find the tangent line to f(x) = x³ − 2x at x = 1. The point is f(1) = 1 − 2 = −1, so (1, −1). The derivative is f′(x) = 3x² − 2, so the slope is f′(1) = 1. The line is y + 1 = 1(x − 1), which simplifies to y = x − 2.
Frequently asked questions
What is a tangent line?
A straight line that touches a curve at one point and has the same slope as the curve there. Its slope is the derivative at that point.
What is the equation of a tangent line?
y − f(a) = f′(a)(x − a). Rearranged, y = f′(a)·x + (f(a) − a·f′(a)).
How do I find the normal line?
The normal line is perpendicular to the tangent, so its slope is −1 ÷ f′(a). Use the same point (a, f(a)).