Derivative from the Definition
The difference quotient, worked out one move at a time . Substitution
gives Say what you would do next before you press the
button. Choosing the move is the skill; doing the algebra is the easy
half.
f (x ) =
at x =
Leave the point blank for the general derivative f ′(x );
give a number for f ′ at that one place.
Write it as you would say it: 2x^2, x^3-2x,
1/x, sqrt(x), x/(x+1).
2x^2
cubic
square root
1/x
x/(x+1)
at a point
Derivative from the Definition
Type a function. The page works out
one move at a time, naming the move in the margin beside the line that made
it. Leave the point blank for f ′(x ); give a number for
f ′ at that place.
Why nothing ever equals 0/0
Substituting h = 0 into the quotient gives 0/0, and you will see
that said in the margin — never on a line. 0/0 is not a number the limit
equals. It is the result of an attempt that failed, which is why the
next move is to rewrite. Writing "
For the same reason
The three moves
Polynomial — multiply out, collect, factor h , cancel.
Everything without an h in it cancels; that is what makes the
rest divisible by h .Rational — common denominator first, then the same thing.Root — multiply top and bottom by the conjugate, which clears the
root by (a − b )(a + b ) = a ² − b ².
Cancelling h is legal because the limit never uses h = 0 — it
only uses values near 0, and at every one of those
h /h = 1. That sentence is the whole justification, and it is
on the line that does the cancelling.
What it will not do
sin, cos, ln and e ˣ are declined , not guessed at: their
difference quotients need a special limit such as
To see the rules applied instead, use
Worked Derivative .
Putting this in a slide or a page
?f=sqrt(x)&embed=true&steps=all&controls=none
gives the finished working with no function box. Leave steps
off and the class advances it a line at a time. See
docs/url-api.md for every parameter.
Keyboard
N Next step
A Show all the working
? This dialog
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