The **tangent line** of a curve at a point *A*, is a line that is tangent, or "touches" the curve, at that point. In effect, *A* belongs to both the figure (the curve) and the tangent line.

## Applications in calculus

Tangent lines are one of the applications of derivatives. The slope of the tangent line of a function *f* of *x* at a point (*a*, *f*(*a*)) is equal to the value of the derivative of *f* at that point:

- $ m = f'(a) $

where *m* is the slope of the tangent line.

The full equation for the tangent line is:

- $ y = f'(a)(x-a) + f(a) $.

The tangent can alternatively be found (as long as the x-coordinate of the point *A* is given and the equation of the curve) by substituting x into the equation to find y. We can then differentiate the equation of the curve, substituting x in to find the gradient. We can then write the standard equation of a straight line using our existing known values as:

- $ y = m (x) + c $

We can then rearange to find c and find the equation of the tangent.