As we recall that the equation of a line that passes through the point  with slope m is:

Hence, the equation of the tangent to the curve y=f(x) at the point  is:

Recall that the slope of the tangent at a point equals the first derivative of the function at the same point.

The normal to the curve with the equation  at the point is that line perpendicular to the tangent line at Thus, if the slope of the tangent at that point is , the slope of the normal will be . Hence, the equation of the normal is:

Example

Find the equation of the tangent and the normal to the curve , at x=2.

Solution

Example

Prove that two curves  are tangent at some point, and find the equation of their tangent line at that point.

Solution