Wikipedia has related information at List of differentiation identities
d d θ tan θ = sec 2 θ {\displaystyle {\frac {d}{d\theta }}\tan \theta =\sec ^{2}\theta }
d d θ cot θ = − csc 2 θ {\displaystyle {\frac {d}{d\theta }}\cot \theta =-\csc ^{2}\theta }
d d θ sec θ = sec θ tan θ {\displaystyle {\frac {d}{d\theta }}\sec \theta =\sec \theta \tan \theta }
d d θ csc θ = − csc θ cot θ {\displaystyle {\frac {d}{d\theta }}\csc \theta =-\csc \theta \cot \theta }
d d x e x = e x {\displaystyle {\frac {d}{dx}}e^{x}=e^{x}}
d d θ sin k θ = k cos k θ {\displaystyle {\frac {d}{d\theta }}\sin k\theta =k\cos k\theta }
d d θ cos k θ = − k sin k θ {\displaystyle {\frac {d}{d\theta }}\cos k\theta =-k\sin k\theta }
d d θ θ = 1 2 θ {\displaystyle {\frac {d}{d\theta }}{\sqrt {\theta }}={\frac {1}{2{\sqrt {\theta }}}}}
d d x e k x = k e k x {\displaystyle {\frac {d}{dx}}e^{kx}=ke^{kx}}
d d θ ln θ = 1 θ {\displaystyle {\frac {d}{d\theta }}\ln \theta ={\frac {1}{\theta }}}
d d θ sin − 1 θ = 1 1 − θ 2 {\displaystyle {\frac {d}{d\theta }}\sin ^{-1}\theta ={\frac {1}{\sqrt {1-\theta ^{2}}}}}
d d θ cos − 1 θ = − 1 1 − θ 2 {\displaystyle {\frac {d}{d\theta }}\cos ^{-1}\theta =-{\frac {1}{\sqrt {1-\theta ^{2}}}}}
d d θ tan − 1 θ = 1 1 + θ 2 {\displaystyle {\frac {d}{d\theta }}\tan ^{-1}\theta ={\frac {1}{1+\theta ^{2}}}}
d d θ e i θ = i e i θ {\displaystyle {\frac {d}{d\theta }}e^{i\theta }=ie^{i\theta }} , where i = − 1 {\displaystyle i={\sqrt {-1}}}
The following deal with the variable, θ {\displaystyle \theta } , and a function, δ {\displaystyle \delta } , of θ {\displaystyle \theta } , and are examples of the chain rule in action.
d d θ sin δ = ( cos δ ) d d θ δ {\displaystyle {\frac {d}{d\theta }}\sin \delta =(\cos \delta ){\frac {d}{d\theta }}\delta }
d d θ δ 4 = ( 4 δ 3 ) d d θ δ {\displaystyle {\frac {d}{d\theta }}\delta ^{4}=(4\delta ^{3}){\frac {d}{d\theta }}\delta }
d d θ θ 3 δ = 3 θ 2 δ + ( θ 3 ) d d θ δ {\displaystyle {\frac {d}{d\theta }}\theta ^{3}\delta =3\theta ^{2}\delta +(\theta ^{3}){\frac {d}{d\theta }}\delta }
d d θ θ 2 δ 2 = 2 θ δ 2 + ( 2 θ 2 δ ) d d θ δ {\displaystyle {\frac {d}{d\theta }}\theta ^{2}\delta ^{2}=2\theta \delta ^{2}+(2\theta ^{2}\delta ){\frac {d}{d\theta }}\delta }
, where 
, and a function, 
, of , and are examples of the chain rule in action.
