Derivative of the product of a function by a constant (possible typo)

In summary, the derivative of the product of a function by a constant is equal to the constant multiplied by the derivative of the function. The constant can be any real number and will not affect the overall derivative. The product rule still applies when a constant is involved and the constant can be treated as a separate function. To find the derivative of a product of a function and a constant, take the derivative of the function as if the constant was not there, and then multiply by the constant. The constant can also be inside the function, in which case the derivative would be found using the chain rule.
  • #1
mcastillo356
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TL;DR Summary
I think there is a typo in this demo
Hi, PF, I think I've found a typo in my textbook. It says:

"In the case of a multiplication by a constant, we've got

$$(Cf)'(x)=\displaystyle\lim_{h \to{0}}{\dfrac{Cf(x+h)-Cf(x)}{h}}=\displaystyle\lim_{h \to{0}}{\dfrac{f(x+h)-f(x)}{h}}=Cf'(x)$$"

My opinion: it should be

$$(Cf)'(x)=\displaystyle\lim_{h \to{0}}{\dfrac{Cf(x+h)-Cf(x)}{h}}=C\displaystyle\lim_{h \to{0}}{\dfrac{f(x+h)-f(x)}{h}}=Cf'(x)$$

Greetings!
 
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  • #2
Definately a typo.
 
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FAQ: Derivative of the product of a function by a constant (possible typo)

What is the product rule for derivatives?

The product rule states that the derivative of a product of two functions is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function.

Can the product rule be applied to a constant multiplied by a function?

Yes, the product rule can be applied to a constant multiplied by a function. The constant can be factored out and treated as a separate function.

Is the derivative of a product of a function by a constant the same as the derivative of the constant multiplied by the function?

Yes, the derivative of a product of a function by a constant is the same as the derivative of the constant multiplied by the function. This is because the order in which the function and the constant are multiplied does not affect the result.

Can the product rule be applied to more than two functions?

Yes, the product rule can be applied to more than two functions. The general rule is that the derivative of a product of n functions is equal to the first function times the derivative of the product of the remaining (n-1) functions, plus the second function times the derivative of the product of the remaining (n-1) functions, and so on.

What is the purpose of the product rule in calculus?

The product rule is an important tool in calculus that allows us to find the derivative of a product of two or more functions. It is particularly useful in solving problems involving rates of change, optimization, and related rates.

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