Tangent line

Summary

Let $f$ be differentiable at $a$. The tangent line to the graph of $f$ at $(a, f(a))$ is the line $y = f(a) + f'(a)(x - a)$. It is the line through this point that best fits the graph near it.

Remarks

The tangent line is the limit of the secant lines through $(a, f(a))$ and $(x, f(x))$ as $x \to a$. The error $f(x) - f(a) - f'(a)(x - a)$ is $o(x - a)$ as $x \to a$. This property characterizes the derivative. For $f(x) = x^2$ at $a = 1$, the tangent line is $y = 2x - 1$.

Explanations

L1 human

The tangent line touches the graph at one point and has the same slope as the graph there. Near the point, the graph and the line are almost the same.