The Derivative
- A function is not differentiable at
if there is a hole, vertical asymptote, or cusp at .
The derivative of a function
Assuming the limit exists. If the limit does not exist, then
Also see The Derivative as a Function
- Lagrange's notation:
, read as "f prime x". Preferred for simple statements and calculations. - Leibniz's notation:
, read as "the derivative of y with respect to x". Preferred for more sophisticated calculations, - Newton's notation:
. Used mostly in physics. - Euler's notation:
. Used when treating the derivative exclusively as an operator. Just a more concise way of writing
The act of calculating a derivative is called differentiation, not deriving.
Geometric Interpretation
The average rate of change of a function
If we let
The Derivative as a Function and Newton's Quotient
If we write the difference
Or, the same "grade 12 definition":
We refer to
Higher-order Derivatives
Since the derivative of a function
In Lagrange's notation, we add more strokes:
In Leibniz's notation, we write:
Differentiability
A function if not differentiable at
- There is a hole at
- There is a vertical asymptote at
- There is a cusp at
, since no unique tangent line will exist
top = 10; bottom = -5;
left = -10; right = 10;
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y = x \{x \ge 0\} | #2d70b3
y = -x \{x < 0\} | #2d70b3
y = 0 | #c74440 | dashed
y = \frac{x}{8} | #c74440 | dashed
y = \frac{-x}{8} | #c74440 | dashed
y = \frac{5x}{8} | #c74440 | dashed
y = \frac{-5x}{8} | #c74440 | dashed