Differentiation is the name for the operation of taking a derivative of a function, and \(\frac{\mathrm{d}}{\mathrm{d}x}\) is called the differential operator. Recall that the derivative of \(f\) with respect to \(x\) is defined as the limit \[ \frac{\mathrm{d}}{\mathrm{d}x}f(x) = f'(x) = \lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}\,. \] Referring to this limit every time we take a derivative would becomes cumbersome. Instead we describe some general “rules” for taking derivatives based on patters we notice based on this definition.
Differentiation is a linear operation; constants factor out and it splits up across sums and differences. I.e. for any constant \(k,\) \[ \frac{\mathrm{d}}{\mathrm{d}x}\biggl(kf(x) \biggr) = k \frac{\mathrm{d}}{\mathrm{d}x}f(x) \qquad\text{ and }\qquad \frac{\mathrm{d}}{\mathrm{d}x}\biggl(f(x) \pm g(x) \biggr) = \frac{\mathrm{d}}{\mathrm{d}x} f(x) \pm \frac{\mathrm{d}}{\mathrm{d}x} g(x) \]
The derivative of any power function has a nice form, a fact that we call the power rule of differentiation: \[ \frac{\mathrm{d}}{\mathrm{d}x}\biggl( x^n \biggr) = n x^{n-1} \]