Www.apcalculusbc.com. is called the derivative of at. We write: “The derivative of f with respect...
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Transcript of Www.apcalculusbc.com. is called the derivative of at. We write: “The derivative of f with respect...
![Page 2: Www.apcalculusbc.com. is called the derivative of at. We write: “The derivative of f with respect to x is …” There are many ways to write the derivative.](https://reader037.fdocuments.net/reader037/viewer/2022110400/56649da85503460f94a948c7/html5/thumbnails/2.jpg)
0
limh
f a h f a
h
is called the derivative of at .f a
We write: 0
limh
f x h f xf x
h
“The derivative of f with respect to x is …”
There are many ways to write the derivative of y f x
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![Page 3: Www.apcalculusbc.com. is called the derivative of at. We write: “The derivative of f with respect to x is …” There are many ways to write the derivative.](https://reader037.fdocuments.net/reader037/viewer/2022110400/56649da85503460f94a948c7/html5/thumbnails/3.jpg)
f x “f prime x” or “the derivative of f with respect to x”
y “y prime”
dy
dx“dee why dee ecks” or “the derivative of y with
respect to x”
df
dx“dee eff dee ecks” or “the derivative of f with
respect to x”
df x
dx“dee dee ecks uv eff uv ecks” or “the derivative
of f of x”( of of )d dx f x
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![Page 4: Www.apcalculusbc.com. is called the derivative of at. We write: “The derivative of f with respect to x is …” There are many ways to write the derivative.](https://reader037.fdocuments.net/reader037/viewer/2022110400/56649da85503460f94a948c7/html5/thumbnails/4.jpg)
dx does not mean d times x !
dy does not mean d times y !
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![Page 5: Www.apcalculusbc.com. is called the derivative of at. We write: “The derivative of f with respect to x is …” There are many ways to write the derivative.](https://reader037.fdocuments.net/reader037/viewer/2022110400/56649da85503460f94a948c7/html5/thumbnails/5.jpg)
dy
dx does not mean !dy dx
(except when it is convenient to think of it as division.)
df
dxdoes not mean !df dx
(except when it is convenient to think of it as division.)
![Page 6: Www.apcalculusbc.com. is called the derivative of at. We write: “The derivative of f with respect to x is …” There are many ways to write the derivative.](https://reader037.fdocuments.net/reader037/viewer/2022110400/56649da85503460f94a948c7/html5/thumbnails/6.jpg)
(except when it is convenient to treat it that way.)
df x
dxdoes not mean times !
d
dx f x
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![Page 7: Www.apcalculusbc.com. is called the derivative of at. We write: “The derivative of f with respect to x is …” There are many ways to write the derivative.](https://reader037.fdocuments.net/reader037/viewer/2022110400/56649da85503460f94a948c7/html5/thumbnails/7.jpg)
In the future, all will become clear.
![Page 8: Www.apcalculusbc.com. is called the derivative of at. We write: “The derivative of f with respect to x is …” There are many ways to write the derivative.](https://reader037.fdocuments.net/reader037/viewer/2022110400/56649da85503460f94a948c7/html5/thumbnails/8.jpg)
y f x
y f x
The derivative is the slope of the original function.
The derivative is defined at the end points of a function on a closed interval.
![Page 9: Www.apcalculusbc.com. is called the derivative of at. We write: “The derivative of f with respect to x is …” There are many ways to write the derivative.](https://reader037.fdocuments.net/reader037/viewer/2022110400/56649da85503460f94a948c7/html5/thumbnails/9.jpg)
2 3y x
2 2
0
3 3limh
x h xy
h
2 2 2
0
2limh
x xh h xy
h
2y x
0lim 2h
y x h
0
![Page 10: Www.apcalculusbc.com. is called the derivative of at. We write: “The derivative of f with respect to x is …” There are many ways to write the derivative.](https://reader037.fdocuments.net/reader037/viewer/2022110400/56649da85503460f94a948c7/html5/thumbnails/10.jpg)
A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points.
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