Constant factor rule in differentiation

# Constant factor rule in differentiation

Discussion
 Ask a question about 'Constant factor rule in differentiation' Start a new discussion about 'Constant factor rule in differentiation' Answer questions from other users Full Discussion Forum

Encyclopedia
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.

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.