Definition of the Derivative
Formal Definition: 
Alternate Definition: 
Example Problem: Find the slope of f(x)
at x = -2 if f(x) = x2 - 3.
Now find the equation of the line tangent to f(x) at (-2,1).
y - 1 = - 4(x + 2)
Characteristics that destroy differentiability:
- Sharp Turns (right-hand limit does not equal left-hand limit)
- Graph y = |x|
- This function is not differentiable at x = 0
- Vertical Tangents
- Graph y = x1/3
- This function is not differentiable at x = 0
because the
slope does not exist there. The tangent is a vertical line.
- Discontinuities
- Graph y = 1/x
- This function is not differentiable at x = 0
because the
function does not exist there.
Problems
