How do you find the equation of a tangent to a curve at a certain point, from the equation of the curve?

First thing you would do is write down the coordinates where the tangent meets the curve. After that you would differenciate the equation (f(x)) of the curve (do dy/dx) in order to get f'(x).Then you would subsitute in the value of x at the point where the tangent meets the curve into f'(x), which is the differenciated equation, which would give you the gradient of the tangent. after that you know that y = MX + C therefore you can subsitute in the values for y (the y coordinate at the point where the tangent and the curve meet), the M (The gradient of the tangent) and X (the x co-ordinate at the point where the tangent meets the curve). After that, all you have to do is rearange the equation to get C and that will give you the eqution of the tangent to the curve at that point.