A derivative is a mathematical operation to find the rate of change of a function.
For a non linear function
. The rate of change of
correspond to change of
is equal to the ratio of change in
over change in
![{\displaystyle {\frac {\Delta f(x)}{\Delta x}}={\frac {\Delta y}{\Delta x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e355c4586822995dece6440c8ad9c9245e601b1e)
Then the Derivative of the function is defined as
![{\displaystyle {\frac {d}{dx}}f(x)=\lim _{\Delta x\to 0}\sum {\frac {\Delta f(x)}{\Delta x}}=\lim _{\Delta x\to 0}\sum {\frac {y}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/412e6fe7f1bf8a9df2ed2e3794639b439417b5ad)
but the derivative must exist uniquely at the point x. Seemingly well-behaved functions might not have derivatives at certain points. As examples,
has no derivative at
;
has two possible results at
(-1 for any value for which
and 1 for any value for which
) On the other side, a function might have no value at
but a derivative of
, for example
at
. The function is undefined at
, but the derivative is 0 at
as for any other value of
.
Practically all rules result, directly or indirectly, from a generalized treatment of the function.
![{\displaystyle {\frac {d}{dx}}\sinh(x)=\cosh(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/555b4c83b6bccb84ebf273ac2d93b852d8bc959d)
![{\displaystyle {\frac {d}{dx}}\cosh(x)=\sinh(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/989042c1c014757193432235013a8dc198595daa)
![{\displaystyle {\frac {d}{dx}}\tanh(x)={\rm {sech}}^{2}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b5b45ed2d7cbec654f165444b2954042bd94661)
![{\displaystyle {\frac {d}{dx}}{\rm {sech}}(x)=-\tanh(x){\rm {sech}}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e3a54d92b890fab7b0c7659ee26a263cbeca789)
![{\displaystyle {\frac {d}{dx}}\coth(x)=-{\rm {csch}}^{2}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd5761a940e46a6f7a723877a84daa69339ec7d0)
![{\displaystyle {\frac {d}{dx}}{\rm {csch}}(x)=-\coth(x){\rm {csch}}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d32f7fb8ce254c2a38ea11188144a14207c699d)
![{\displaystyle {\frac {d}{dx}}{\rm {arcsinh}}(x)={\frac {1}{\sqrt {x^{2}+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cd19eefc8bdc2b076278e7e3ee4ecc76c02ea03)
![{\displaystyle {\frac {d}{dx}}{\rm {arccosh}}(x)=-{\frac {1}{\sqrt {x^{2}-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd63be477134bab31b130410bfcf8b4088d142a)
![{\displaystyle {\frac {d}{dx}}{\rm {arctanh}}(x)={\frac {1}{1-x^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d23777709ea8b9f2ab26fc1c3306748ab67a1c38)
![{\displaystyle {\frac {d}{dx}}{\rm {arcsech}}(x)={\frac {1}{x{\sqrt {1-x^{2}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59af5ec29bcf703bf51f968247f7193814e1c556)
![{\displaystyle {\frac {d}{dx}}{\rm {arccoth}}(x)=-{\frac {1}{1-x^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6285df4e30ef6065f0ecddcf96fe87337476b7c)
![{\displaystyle {\frac {d}{dx}}{\rm {arccsch}}(x)=-{\frac {1}{|x|{\sqrt {1+x^{2}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ede1595222473ecba5dfefb327b5b3ab30ce5cfb)
- Derivative
- Table_of_derivatives