Equations · Calculus and analysis
Definition of the derivative
The instantaneous rate of change of a function, as the limit of the slope over a shrinking interval.
f′(x) = lim_{h→0} (f(x + h) − f(x)) / h
Aethel says
f prime of x equals the limit as h goes to 0 of the quantity f of the quantity x plus h, minus f of x, over h
What it says
The fraction is the slope of the straight line through two points of the graph, at x and at x plus h. As h shrinks towards zero, the line turns into the tangent at x, and its slope is the derivative.
The derivative of position is velocity; the derivative of velocity is acceleration. Every rule of differentiation is proved from this definition.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| f′(x) | f prime of x | the derivative of f at x |
| lim_{h→0} | the limit as h goes to 0 of | the value the expression approaches as h gets ever closer to zero |
| h | h | a small step along the x axis |
Where you meet it
The first weeks of every calculus course.
Questions
How do you say Definition of the derivative out loud?
f′(x) = lim_{h→0} (f(x + h) − f(x)) / h is read: “f prime of x equals the limit as h goes to 0 of the quantity f of the quantity x plus h, minus f of x, over h.” That is exactly what Aethel says when it meets the formula in a book.
What does f′(x) mean in Definition of the derivative?
f′(x) (said “f prime of x”) is the derivative of f at x.
What does lim_{h→0} mean in Definition of the derivative?
lim_{h→0} (said “the limit as h goes to 0 of”) is the value the expression approaches as h gets ever closer to zero.
Can I listen to the equations in my own textbook like this?
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