Constant multiple rule

In calculus, the constant factor rule in differentiation allows one to take constants outside a derivative and concentrate on differentiating the function of x itself. This is a part of the linearity of differentiation.

Consider a differentiable function

g(x)=k\cdot f(x).

where k is a constant.

Use the formula for differentiation from first principles to obtain:

{\begin{aligned}g'(x)&=\lim _{h\to 0}{\frac {g(x+h)-g(x)}{h}}\\&=\lim _{h\to 0}{\frac {k\cdot f(x+h)-k\cdot f(x)}{h}}\\&=\lim _{h\to 0}{\frac {k(f(x+h)-f(x))}{h}}\\&=k\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}\\&=k\cdot f'(x).\end{aligned}}

This is the statement of the constant factor rule in differentiation, in Lagrange's notation for differentiation.

{\frac {d(k\cdot f(x))}{dx}}=k\cdot {\frac {d(f(x))}{dx}}.

If we put k = −1 in the constant factor rule for differentiation, we have:

{\frac {d(-y)}{dx}}=-{\frac {dy}{dx}}.

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