Example 1. The slope of the tangent of the curve Y = 2x 2+3 at the point (-1, 5) is _ _ _ _ _ _ _ _ _ _ _ _?

Derive directly to get y'=4x, and substitute x=- 1 to get y'=-4, so the slope is -4.

Example 2. The tangent equation of curve y = x 3+1 at (1,2) is _ _ _ _ _ _ _ _ _ _ _?

Take the derivative first, y'=3x 2, and substitute x= 1 to get y' = 3.

Let the tangent equation be y=3x+b, and 3 is the slope just obtained, because the point (1, 2) passes through both the original straight line and the tangent line, and b=- 1 is obtained by substitution.

So the tangent equation is y=3x- 1.