The Tangent Line
The derivative has a beautiful geometric meaning: it's the slope of the tangent line to a curve at a point.
A tangent line just "touches" the curve at one point - it doesn't cross through it (locally). The derivative tells us the slope of this line.
Derivative as Tangent Slope
The derivative at a point equals the slope of the tangent line
Illustration: tangent-line-derivative
Tangent Line Equation
Tangent Line
y - y_0 = f'(x_0)(x - x_0)
The tangent line at point (x_0, y_0) has slope f'(x_0)
- x_0:
- The x-value of the point
- y_0:
- f(x_0), the y-value
- f'(x_0):
- The derivative at x_0 (the slope)
Finding a Tangent Line
Medium
Find the tangent line to f(x) = x^2 at x = 3
1
Find the point
y_0 = f(3) = 3^2 = 9
Point: (3, 9)
2
Find the slope
f'(x) = 2x
f'(3) = 2(3) = 6
3
Write tangent line
y - 9 = 6(x - 3)
y = 6x - 18 + 9
y = 6x - 9
Answer:
y = 6x - 9
Practice
1. Find the slope of the tangent to f(x) = x^3 at x = 2
f'(x) = 3x^2, so f'(2) = 3(4) = 12
Key Takeaways
The derivative = slope of tangent line
Tangent line equation: y - y_0 = m(x - x_0)
Find point, find slope, write equation