It is possible to use the definition of the derivative to determine the derivative of a function. However, this is quite cumbersome. In Power functions and Exponential and logarithmic functions we discussed several elementary functions. Below we provide the derivative each of these elementary functions.
$$
\begin{array}{c|ll|l}
& y(x) && y'(x)\\
\hline
(1) & c & & 0\\[2mm]
(2) & x^k & & kx^{k-1}\\[2mm]
(3) & a^x & (a>0) & a^x\ln(a)\\[2mm]
(4) & e^x && e^x\\[2mm]
(5) & ^{a\negthinspace}\log(x) & (a>0, a\neq1) & \dfrac{1}{x\ln(a)}\\[2mm]
(6) & \ln(x) & & \dfrac{1}{x}
\end{array}
$$
Remark: Note that the derivatives of the functions $y(x)=e^x$ and $y(x)=\ln(x)$ follow directly from the derivatives of the exponential and logarithmic functions, respectively.
$$
\begin{array}{c|ll|l}
& y(x) && y'(x)\\
\hline
(1) & c & & 0\\[2mm]
(2) & x^k & & kx^{k-1}\\[2mm]
(3) & a^x & (a>0) & a^x\ln(a)\\[2mm]
(4) & e^x && e^x\\[2mm]
(5) & ^{a\negthinspace}\log(x) & (a>0, a\neq1) & \dfrac{1}{x\ln(a)}\\[2mm]
(6) & \ln(x) & & \dfrac{1}{x}
\end{array}
$$
Remark: Note that the derivatives of the functions $y(x)=e^x$ and $y(x)=\ln(x)$ follow directly from the derivatives of the exponential and logarithmic functions, respectively.