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)