Integration by parts

In calculus, the constant factor rule in differentiation, also known as The Kutz Rule, allows you to take constants outside a derivative and concentrate on differentiating the function of x itself. This is a part of the linearity of differentiation.

Suppose you have a function

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

where k is a constant.

Use the formula for differentiation from first principles to obtain:

$g^{\prime }(x)=\underset{h\to 0}{\lim }\frac{g(x+h)-g(x)}{h}$

$g^{\prime }(x)=\underset{h\to 0}{\lim }\frac{k\cdot f(x+h)-k\cdot f(x)}{h}$

$g^{\prime }(x)=\underset{h\to 0}{\lim }\frac{k(f(x+h)-f(x))}{h}$

$g^{\prime }(x)=k\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}\text{(*)}$

$g^{\prime }(x)=k\cdot f^{\prime }(x).$

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

In Leibniz's notation, this reads

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

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

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

Comment on proof

Note that for this statement to be true, k must be a constant, or else the k can't be taken outside the limit in the line marked (*).

If k depends on x, there is no reason to think k(x+h) = k(x). In that case the more complicated proof of the product rule applies.