How do I find the equation of a tangent to a given point on a curve?

Let's take an example: Curve: y = x^2 Point: (2,4)

The equation of the tangent is going to be of the form y = mx + c (standard equation of a straight line), where m is the gradient of the line and c is the intersection on the y axis. Once we find 'm' and 'c' we have the answer.

Step 1: The first thing to do is find 'm', the gradient. To do this we use differentiation, this gives us the rate of change in y against x - i.e. the gradient. For y=x^2, dy/dx = 2x. This is the general gradient for the curve, we need the gradient at our specific point (2,4) so we plug that in to our equation: dy/dx = 2x = 2*2 = 4 = 'm'. So we know y = 4x + c.

Step 2: We have 'm' now so just need to find 'c'. We know our tangent line is y = 4x + c so plugging in our point (2,4) which we know is on the line, we get 4 = 4*2 + c = 8 + c, i.e. c = -4.

Step 3: Pulling all this together, we can say that the answer is y = 4x - 4 or y = 4(x-1)