The constant factor rule and the sum and difference rule

Differentiation, derivatives and Taylor approximations: Rules for differentiating functions

The constant factor rule and the sum and difference rule

The following rules are used in differentiating a function which is

a multiple of another differentiable function;
the sum or difference of two differentiable functions.

Constant factor rule, sum rule and difference rule for differentiation

If \(f\) and \(g\) are differentiable at \(x\) we have the following rules: \[\begin{aligned}
\bigl(c\cdot f(x)\bigr)' &= c\cdot f'(x)\quad \text{for any constant }c\\ \\
\bigl(f(x)+g(x)\bigr)' &= f'(x)+g'(x)\\ \\
\bigl(f(x)-g(x)\bigr)' &= f'(x)-g'(x)
\end{aligned}\]

Examples

\[\begin{aligned}\\
\bigl(2x^3\bigr)' &= 2\cdot(x^3)' =2\cdot(3\cdot x^2)=6x^2 \\ \\
\bigl(x^2+x^3\bigr)' &= (x^2)'+(x^3)'=2x+3x^2\\ \\
\bigl(x^2-x^3\bigr)' &= (x^2)'-(x^3)'=2x-3x^2
\end{aligned}\]

For math enthusiasts we present proofs of the above rules for differentiation.

For \(F(x)=c\cdot f(x)\) , \(s(x)=f(x)+g(x)\) and \(v(x)=f(x)-g(x)\) we have: \[\begin{aligned}\frac{{\vartriangle}F}{{\vartriangle}x}
&= \frac{F(x+{\vartriangle}x)-F(x)}{{\vartriangle}x}\\ \\
&= \frac{c\cdot f(x+{\vartriangle}x)-c\cdot f(x)}{{\vartriangle}x}\\ \\
&= c\cdot \frac{ f(x+{\vartriangle}x)- f(x)}{{\vartriangle}x}\\ \\
&\to c\cdot f'(x)\quad\text{when }{{\vartriangle}x}\to 0.\\ \\ \\ \\ \\ \\
\frac{{\vartriangle}s}{{\vartriangle}x}&= \frac{ f(x+{\vartriangle}x)+g(x+{\vartriangle}x)- \bigl(f(x)+g(x)\bigr)}{{\vartriangle}x}\\ \\
&= \frac{ f(x+{\vartriangle}x)-f(x)}{{\vartriangle}x} +\frac{g(x+{\vartriangle}x)- g(x)}{{\vartriangle}x}\\ \\
&\to f'(x)+g'(x)\quad\text{when }{{\vartriangle}x}\to 0.\\ \\ \\ \\ \\ \\
\frac{{\vartriangle}v}{{\vartriangle}x}&= \frac{ f(x+{\vartriangle}x)-g(x+{\vartriangle}x)- \bigl(f(x)-g(x)\bigr)}{{\vartriangle}x}\\ \\
&= \frac{ f(x+{\vartriangle}x)-f(x)}{{\vartriangle}x} -\frac{g(x+{\vartriangle}x)- g(x)}{{\vartriangle}x}\\ \\
&\to f'(x)-g'(x)\quad\text{when }{{\vartriangle}x}\to 0.\\ \\ \\ \\ \end{aligned}\]

The last rule for differentiation is somewhat redundant since it already follows from the first two: \[\begin{aligned}\bigl(f(x)-g(x)\bigr)'&=\biggl(f(x)+ \bigl(-1\cdot g(x)\bigr)\biggr)'\\ \\ &=f'(x)+ \bigl(-1\cdot g(x)\bigr)'\qquad\blue{\text{sum rule}}\\ \\ &=f'(x)+ \bigl(-1\cdot g'(x)\bigr)\qquad\blue{\text{constant factor rule}}\\ \\ &=f'(x)-g'(x)\end{aligned}\]

We call the above rules for differentiation the constant factor rule, the sum rule, and the difference rule, respectively.

For those who lose the overview due to the many letters in the calculation rules or who cannot remember the rules well in this way: one can omit the independent variable in the functions, only use calculus with differentiable functions, and write the calculation rules as follows.

Constant factor rule, sum rule and difference rule for differentiation in short notation For differentiable functions \(f\) and \(g\) we have:\[\begin{aligned}
\bigl(c\cdot f\bigr)' &= c\cdot f'\quad \text{for any constant }c\\ \\
\bigl(f+g\bigr)' &= f'+g'\\ \\
\bigl(f-g\bigr)' &= f'-g'
\end{aligned}\]