Let
be a subset,
a point, and
-
functions
which are
differentiable
in
. Then the following rules for differentiability holds.
- The sum
is differentiable in
, with
-
![{\displaystyle {}(f+g)'(a)=f'(a)+g'(a)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa5b3e69f3228865312bddeba440ba8fd73c0b02)
- The product
is differentiable in
, with
-
![{\displaystyle {}(f\cdot g)'(a)=f'(a)g(a)+f(a)g'(a)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e53ea6380874689593f00937460e325ba6cf838b)
- For
,
also
is differentiable in
, with
-
![{\displaystyle {}(cf)'(a)=cf'(a)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95f3bfb50134f47416838494509c63a7116b5cb2)
- If
has no zero in
, then
is differentiable in
, with
-
![{\displaystyle {}{\left({\frac {1}{g}}\right)}'(a)={\frac {-g'(a)}{(g(a))^{2}}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8bf8c052575ce91bbb24b7827e4a9eeab518975)
- If
has no zero in
, then
is differentiable in
, with
-
![{\displaystyle {}{\left({\frac {f}{g}}\right)}'(a)={\frac {f'(a)g(a)-f(a)g'(a)}{(g(a))^{2}}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d6aecd68a5e0fcd0f300f4557ade04b8b023d64)