In the following formulas the angle u is supposed to be expressed in circular measure.
![{\displaystyle {\operatorname {d} \over \operatorname {d} x}\sin u=\cos u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1df24ac8f023594de7670fb31faae9b4dc0252bf)
![{\displaystyle {\operatorname {d} \over \operatorname {d} x}\cos u=-\sin u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d03c64ec87f1f8e9299548f27c20b8e7313e8e3)
![{\displaystyle {\operatorname {d} \over \operatorname {d} x}\tan u=\sec ^{2}u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19f5af2298eb509ac5ddc466a1c48dbfe8d3c3f0)
![{\displaystyle {\operatorname {d} \over \operatorname {d} x}\cot u=-\csc ^{2}u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d75469fd0ea27c6d480012e7eb07508c0300a68)
![{\displaystyle {\operatorname {d} \over \operatorname {d} x}\sec u=\sec u\tan u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08326971fab235318c155b77f3bf8fb93a260a95)
![{\displaystyle {\operatorname {d} \over \operatorname {d} x}\csc u=-\csc u\cot u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3e9566904c04a62507330da5a1957fccf25b1de)
Proof for the derivative of ![{\displaystyle \sin u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a12840715005f2cf490e983fe03dc56e6aa26df)
Let
,
then
;
therefore
;
In Trigonometry,
![{\displaystyle \sin A-\sin B=2\sin {1 \over 2}(A-B)\cos {1 \over 2}(A+B)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/590b298a481aae6469ec11aec0144401f11bc210)
If we substitute
and
,
we have
![{\displaystyle \Delta y=2\cos \left(u+{\Delta u \over 2}\right)\ sin{\Delta u \over 2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dea0eab61f76ca6d6b382f499227239733dee8e)
Hence
![{\displaystyle {\Delta y \over \Delta x}=\cos \left(u+{\Delta u \over 2}\right){\sin {\Delta u \over 2} \over {\Delta u \over 2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6978b7962a1d19af170c8d048eac1fe816a685a5)
When
approaches zero,
likewise approaches zero, and as
is in circular measure, the limit of
![{\displaystyle {\sin {\Delta u \over 2} \over {\Delta u \over 2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98705052f02edcac115af30bfb53e9cc70511ac4)
Hence
![{\displaystyle {\operatorname {d} y \over \operatorname {d} x}=\cos u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc5984eef035aff4f224efc336bdd0b02de6fa78)
Proof for ![{\displaystyle {\operatorname {d} \over \operatorname {d} x}\cos u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e96b97b413cd7bfcf7f3bfd185299db1c600a500)
![{\displaystyle {\operatorname {d} \over \operatorname {d} x}\cos u=\sin u\left(-{\operatorname {d} u \over \operatorname {d} x}\right)=-\sin u{\operatorname {d} u \over \operatorname {d} x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a586d80b470d02b9c5dfeccea99cd9438f171ce)
Proof for ![{\displaystyle {\operatorname {d} \over \operatorname {d} x}\tan u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1685a72d8dde146f2b27d96634d648beef52f59)
Since ![{\displaystyle \tan u={\sin u \over \cos u}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/483aa686dd08f147102a6a7b810a89b2e7d984f5)
![{\displaystyle {\operatorname {d} \over \operatorname {d} x}\tan u={\cos u{\operatorname {d} \over \operatorname {d} x}\sin u-\sin u{\operatorname {d} \over \operatorname {d} x}\cos u \over \cos ^{2}u}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ea00a2ae81a31778164d47fc826748fe84e3455)
![{\displaystyle ={\cos ^{2}u{\operatorname {d} u \over \operatorname {d} x}+\sin ^{2}u{\operatorname {d} u \over \operatorname {d} x} \over \cos ^{2}u}={\operatorname {d} u \over \cos ^{2}u}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9056571da5872e5ae26603a39b57dc6ebf504d9)
![{\displaystyle =\sec ^{2}u{\operatorname {d} u \over \operatorname {d} x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57eea2253c997c2aaab0669414a245961cf8bde2)
Proof for ![{\displaystyle {\operatorname {d} \over \operatorname {d} x}\sec u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fad7e3afced589a1f8d5b883ff2b7378062376f)
![{\displaystyle {\operatorname {d} \over \operatorname {d} x}\tan u={\cos u{\operatorname {d} \over {d}x}\sin u-\sin u{\operatorname {d} \over \operatorname {d} x}\cos u \over \cos ^{2}u}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2acc313047b4d4b8095e0b57912edda1492d33d)
![{\displaystyle ={\cos ^{2}{\operatorname {d} u \over \operatorname {d} x}+\sin ^{2}u{\operatorname {d} u \over \operatorname {d} x} \over \cos ^{2}u}={{\operatorname {d} u \over \operatorname {d} x} \over \cos ^{2}u}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cc78278b92b5b8b7b999cc62865608cf66135d5)
![{\displaystyle =\sec ^{2}u{\operatorname {d} u \over \operatorname {d} x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57eea2253c997c2aaab0669414a245961cf8bde2)