The slope of a tangent line to a curve can be found using limits.

The tangent line to a curve is the limit of the secant lines as point Q approaches point P or as Dx 0.

**Slope of a secant line through the points (x,y) and (x+ Dx, y+ Dy)**

(divide by Dx because Dy/Dx is the slope) | |

(slope of the secant line) |

The tangent line to a curve is the limit of secant lines as point Q approaches
point P or as Dx 0.

**Slope of the tangent line to any point on the curve**

**Slope of the tangent line at a specific point**

At the point (2,5) the slope would equal 2(2) = 4

So 4 is the slope of the tangent line to the curve y = x² + 1 at the point (2,5)

The limiting value of the ratio Dy/Dx is the 1st derivative of the function

(Definition of a derivative)