The first derivative \(f'(x)\) is the gradient of the tangent to the graph of \(y=f(x)\) at a general point \((x,y)\).

Equation of tangents

Using the equation of straight lines \(y-y_1=m(x-x_1) \), the equation of a tangent to graph at \((x_1,y_1)\) can be found by using the gradient \(m = f'(x_1)\) and the values of the \(x_1\) and \(y_1\) coordinates.

Equation of normals

Similar to the above, except the gradient of the normal is the negative reciprocal of the gradient of the tangent, i.e. \(m = \dfrac{-1}{f'(x_1)} \).