Rules of differentiation

Introduction: By the use of the Derivatives of elementary functions we are able to determine the derivative of several functions. However, what to do when the function is the sum or product of elementary functions? Or even more difficult function prescriptions? In that case we can use the rules of differentiation as presented below.

Scalar product rule:
If $y(x)=cu(x)$, then $y'(x) = cu'(x)$.

Sum rule:
If $y(x) = u(x) + v(x)$, then $y'(x) = u'(x) + v'(x)$.

Product rule:
If $y(x) = u(x)v(x)$, then $y'(x) = u'(x)v(x) + u(x)v'(x)$.

Quotient rule:
If $y(x) = \dfrac{u(x)}{v(x)}$, then $y'(x) = \dfrac{u'(x)v(x) - u(x)v'(x)}{\big(v(x)\big)^2}$.

The rules of differentiation will be treated one by one in the sequel.