Derivatives appear often in the analysis of electric circuits, particularly when studying circuits in the time domain that include capacitors and inductors. In addition, derivatives also appear when discussing 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 the time domain (particularly when they include capacitors and inductors). In addition, integrals appear in concepts such as energy 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 $$\begin{equation} \int_a^b f(t) g'(t) dt = \left[f(t)g(t)\right]_a^b - \int_a^b f'(t)g(t) dt \end{equation}$$