Constant factor rule in differentiation

In calculus, the constant factor rule in differentiation, also known as The Kutz Rule, allows you to take constants

Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

 outside a derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 and concentrate on differentiating

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 the function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 of x itself. This is a part of the linearity of differentiation

Linearity of differentiation

In mathematics, the linearity of differentiation is a most fundamental property of the derivative, in differential calculus. It follows from the sum rule in differentiation and the constant factor rule in differentiation...

.

Suppose you have a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

where k is a constant.

Use the formula for differentiation from first principles to obtain:


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

In Leibniz's notation, this reads


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

Comment on proof

Note that for this statement to be true, k must be a constant

Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

, or else the k can't be taken outside the limit

Limit of a function

In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

 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

Product rule

In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...

applies.