Tangents and Normals
Finding a tangent's slope
The slope of the tangent to y=f(x) at a specific point is just f′(x) evaluated at that point's x-value.
Worked example: a cubic curve
Find the slope of the tangent to y=x3−x at x=2.
Solution: y′=3x2−1. At x=2:
y′(2)=3(4)−1=11Worked example: a rational function
Find the slope of the tangent to y=x−2x−1 (for x=2) at x=10.
Solution: By the quotient rule:
y′=(x−2)2(1)(x−2)−(x−1)(1)=(x−2)2−1At x=10:
y′(10)=(10−2)2−1=−641Continue learning
The normal line at the same point is perpendicular to the tangent, and its slope follows directly from the tangent slope found here.