Tangent line

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.