4.03. THEOREMS OF DIFFERENTIATION
1. If c is a constant, then \(
\frac{d}
{{dx}}\left( c \right) = 0
\)
2. If y = cu, where c is a constant and u is a function of x, then \(
\frac{{dy}}
{{dx}} = \frac{d}
{{dx}}\left( {cu} \right) = c\frac{{du}}
{{dx}}
\)
3. If \(
y = u \pm v \pm w,
\) where u, are functions of x, then \(
\frac{{dy}}
{{dx}} = \frac{d}
{{dx}}\left( {u \pm \upsilon \pm w} \right) = \frac{{du}}
{{dx}} \pm \frac{{d\upsilon }}
{{dx}} \pm \frac{{dw}}
{{dx}}
\)
4. If y = uv, where u and v are function of x, then \(
\frac{{dy}}
{{dx}} = \frac{d}
{{dx}}\left( {u\upsilon } \right) = u\frac{{d\upsilon }}
{{dx}} + \upsilon \frac{{du}}
{{dx}}
\)
5. If \(
y = \frac{u}
{\upsilon },
\) where u and v are functions of x, then \(
\frac{{dy}}
{{dx}} = \frac{d}
{{dx}}\left( {\frac{u}
{\upsilon }} \right) = \frac{{\upsilon \frac{{du}}
{{dx}} - u\frac{{d\upsilon }}
{{dx}}}}
{{\upsilon ^2 }}
\)
6. If \(
y = x^n
\), where n is a real number, then \(
\frac{{dy}}
{{dx}} = \frac{d}
{{dx}}\left( {x^n } \right) = nx^{n - 1}
\)