Formulas
(cumulative listing — formulas derived in exercises are linked to those exercises)

Exponential Functions

\(\frac{d}{dt}e^t = e^t\), where \(e\) is the natural base, \(2.71828...\)

\(\frac{d}{dt}e^{kt}=ke^{kt}\), for any constant \(k\).

\(\frac{d}{dt}b^t=[\ln b] b^t\), for any constant \(b\).

Power Rule

\(\frac{d}{dt}t^r=rt^{r-1}\), for any real constant \(r\).

Natural logarithm function

\(\frac{d}{dt}\ln t=\frac{1}{t}\)

Trigonometric Functions

\(\frac{d}{dt}\sin t = \cos t\)	\(\frac{d}{dt}\sec t = \tan t \sec t\) (5.5 Pr. 1)
\(\frac{d}{dt}\cos t = -\sin t\)	\(\frac{d}{dt}\csc t = -\cot t \csc t\) (5.5 Pr. 2)
\(\frac{d}{dt}\tan t = \sec^2 t\)	\(\frac{d}{dt}\cot t = -\csc^2 t\)

Inverse Trigonometric Functions

\(\frac{d}{dt}\sin^{-1\,} t = \frac{1}{\sqrt{1-t^2}}\)	\(\frac{d}{dt}\sec^{-1}\, t = \frac{1}{t \sqrt{t^2-1}}\) (5.5 Pr. 6)
\(\frac{d}{dt}\cos^{-1}\, t = \frac{-1}{\sqrt{1-t^2}}\)	\(\frac{d}{dt}\csc^{-1}\, t = \frac{-1}{t \sqrt{t^2-1}}\) (5.5 Pr. 7)
\(\frac{d}{dt}\tan^{-1}\, t = \frac{1}{1+t^2}\)	\(\frac{d}{dt}\cot^{-1}\, t = \frac{-1}{1+t^2}\) (5.5 Pr. 5)

Constant Multiple Rule

\(\frac{d}{dt}Af(t)=A\frac{d}{dt}f(t)\), where \(A\) is any constant.

Sum Rule

\(\frac{d}{dt}[f(t)+g(t)]=\frac{d}{dt}f(t)+\frac{d}{dt}g(t)\)

Product Rule

\(\frac{d}{dt}[g(t)h(t)] = g(t)\frac{d}{dt}h(t)+h(t)\frac{d}{dt}g(t)\)

Quotient Rule (4.7 Problem 10)

\(\frac{d}{dt} \frac{g(t)}{h(t)} = \frac{h(t)g'(t) - g(t)h'(t)}{[h(t)]^2}\), or \(\frac{d}{dt} \frac{u}{v} = \frac{v \frac{du}{dt} -u \frac{dv}{dt}}{v^2}\)

Inverse Function Rule

\(\frac{dy}{dt}=\frac{1}{dt/dy}\)

Chain Rule

\(\frac{dy}{dt}=\frac{dy}{du}\frac{du}{dt}\)

Special case of the Chain Rule

\(\frac{d}{dt}f(kt)=k\frac{d}{du}f(u)\),

where \(u=kt\) and \(k\) is any constant.

Combinations of the Chain Rule with specific function rules

\(\frac{d}{dt}[f(t)]^r=r[f(t)]^{r-1}\frac{d}{dt}f(t)\), for any real constant \(r\).

\(\frac{d}{dt}e^{f(t)}=e^{f(t)}\frac{d}{dt}f(t)\)

\(\frac{d}{dt}\ln f(t) = \frac{1}{f(t)}\frac{d}{dt}f(t)\)

\(\frac{d}{dt}\sin f(t) = \cos f(t) \frac{d}{dt} f(t)\)

\(\frac{d}{dt}\cos f(t) = -\sin f(t) \frac{d}{dt} f(t)\)

\(\frac{d}{dt}\tan f(t) = \sec^2 f(t) \frac{d}{dt} f(t)\)

\(\frac{d}{dt}\sin^{-1} f(t) = \frac{1}{\sqrt{1-f(t)^2}} \frac{d}{dt} f(t)\)

\(\frac{d}{dt}\tan^{-1} f(t) = \frac{1}{1+f(t)^2} \frac{d}{dt} f(t)\)