Tangent and normal lines: Survey of methods

Assume that f is a function that is differentiable at a point a.

The tangent line to the graph of f at a is given by

The normal line to the graph of f at a is given by

Note: Some people prefer a different form since it might be easier to remember,

y - f (a) = f ′(a)⋅(x - a)

for the tangent line and analogously for the normal line. For details see Tangent line in Theory - Applications.

Example: Find the tangent and normal line to the graph of f (x) = sin(x) at x = 0.

Solution: We find f ′(x) = cos(x), therefore f (0) = 0 and f ′(0) = 1. Thus we get the tangent line

y = 1⋅(x - 0) + 0 = x

and the normal line

y = −1⋅(x - 0) + 0 = −x.

For other examples see Tangent line in Theory - Applications and Solved Problems - Applications.

One interesting application of tangent lines is approximation of functions, see Approximation in Theory - Applications.