Derivatives appear often in the analysis of electric circuits, particularly when studying circuits in time-domain and those circuits include capacitors and inductors. In addition, derivatives also appear when discussing about concepts such as power and current. Below is a list of the most common derivatives used in electric circuits. $$\begin{equation} \frac{d}{dt}t^a = at^{a-1} \end{equation}$$ $$\begin{equation} \frac{d}{dt} e^{a t} = a e^{a t} \end{equation}$$ $$\begin{equation} \frac{d}{dt}a^t = a^t \cdot \ln{a} \end{equation}$$ $$\begin{equation} \frac{d}{dt}\ln{a t} = \frac{a}{t}\end{equation}$$ $$\begin{equation} \frac{d}{dt}\log_a{t} = \frac{1}{t \ln{a}}\end{equation}$$ $$\begin{equation} \frac{d}{dt}\sin{t} = \cos{t}\end{equation}$$ $$\begin{equation} \frac{d}{dt}\cos{t} = -\sin{t}\end{equation}$$

Derivatives of combined functions $$\begin{equation} \frac{d}{dt}\left[a f(t) + b g(t)\right] = a f'(t) + b g'(t)\end{equation}$$ $$\begin{equation} \frac{d}{dt}\left[f(t) \cdot g(t)\right] = f(t) g'(t) + f'(t) g(t)\end{equation}$$ $$\begin{equation} \frac{d}{dt}\frac{f(t)}{g(t)} = \frac{g(t) f'(t) - f(t) g'(t)}{g^2(t)}\end{equation}$$ $$\begin{equation} \frac{d}{dt} f(g(t)) = f'(g(t)) \cdot g'(t)\end{equation}$$

Like derivatives, integrals appear in the analysis of electric circuits in time-domain (particularly when then include capacitors and inductors). In addition, they appear when concepts such as ennergy and charge. Below is a list of a few common integrals used in electric circuits. $$\begin{equation} \int_a^b t^a dt = \frac{t^{a+1}}{a+1} \end{equation}$$ $$\begin{equation} \int_a^b e^{a t} dt = \frac{e^{a t}}{a} \end{equation}$$ $$\begin{equation} \int_a^b a^t dt = \frac{a^t}{\ln a} \end{equation}$$ $$\begin{equation} \int_a^b \frac{dt}{t} = \ln |t| \end{equation}$$ $$\begin{equation} \int_a^b \sin{t}\,dt = -\cos{t} \end{equation}$$ $$\begin{equation} \int_a^b \cos{t}\,dt = \sin{t} \end{equation}$$

Combined functions $$\begin{equation} \int_a^b \left[c f(t) + d g(t)\right] dt = c \int_a^b f(t)dt + d \int_a^b g(t)dt\end{equation}$$ Integration by parts $$\begin{equation} \int_a^b f(t) g'(t) dt = \left[f(t)g(t)\right]_a^b - \int_a^b f'(t)g(t) dt \end{equation}$$