# Derivatives of constants

this is just a general question, one that keeps coming up everytime i take a test or quiz.

I understand that the derivative of a constant is zero, but I dont understand when you use that zero to multiply or divide in a problem, and when you just ignore that zero.

For example x^2/3
Would that be zero? or 2x?
Because I know in some problems you just ignore the constant when it is alone, but I'm not sure about when it is a variable.

Help is appreciated, I have a test tomorrow
Thanks

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TD
Homework Helper
The derivative of a constant term is 0, in your case the term isn't constant. The "3" you're probably referring to is a factor which belongs to the term "x²/3".

Since the derivative is lineair, you can put the factor up front and get:

$$\left( {\frac{{x^2 }} {3}} \right)^\prime = \frac{1} {3}\left( {x^2 } \right)^\prime = \frac{{2x}} {3}$$

Doc Al
Mentor
If y = a [a constant by itself], then $dy/dx = 0$.
If y = a f(x) [a constant times a function], then $dy/dx = a (df/dx)$

So if $y = (1/3)x^2$, then $dy/dx = (1/3)(2x) = (2/3)x$

Hurkyl
Staff Emeritus
Gold Member
For example x^2/3
Try applying the division rule to this. (of course, it's better if you remember the scalar multiplication rule, but since this is a sticky point for you, try doing it with the division rule and see if you're happy)

HallsofIvy
Homework Helper
I don't think that is his problem, Hurkyl!

Because (f+ g)'= f'+ g', (f+ Constant)'= f'+ Constant'= f'+ 0= f' so you can "ignore" the constant. (Notice the quotes- you are not IGNORING it! You are THINKING about it, recognizing that its derivative is a constant and so "adding 0".

Because (fg)'= f'g+ fg', (Constant*g)'= (Constant)'g+ (Constant)g'= 0*g+ Constant*g'. The derivative of a constant times a function is the constant times the derivative of the function. I guess you could say you are "ignoring" the constant here but that's why I don't like that word! THINK about what you are doing!

Surely you know that every function involves some constants (even if they are 1). You can't just "ignore" them.

The two rules I mentioned above: (f+ Constant)'= f' and (Constant*f)'= Constant*f' are as close as you can come to "ignoring" the Constant!

Hurkyl
Staff Emeritus
Gold Member
I'm hoping he'll appreciate what's going on better if he does the division rule and sees the constant resurface at the end. In effect, I'm hoping he'll work out for himself exactly what you told him about the product rule.

hotrocks007 said:
For example x^2/3
i'm not seeing a constant there. a quick mental calculation gives $$\frac{d}{dx}f(x)=\frac{2}{3}x^{-\frac{1}{3}}$$

Doc Al
Mentor
i'm not seeing a constant there.
It depends on what the OP meant by "x^2/3": either (x^2)/3 or x^(2/3). (From the context I assumed he meant the former.)

that's why you use $$\LaTeX$$

Hurkyl
Staff Emeritus
Gold Member
And, of course, the strict interpretation x^2/3 is (x^2)/3.

Doc Al
Mentor
Hurkyl said:
And, of course, the strict interpretation x^2/3 is (x^2)/3.
And that's even true in Latex: x^2/3 = $x^2/3$.

then it's $$\frac{d}{dx}f(x)=\frac{2}{3}x$$

HallsofIvy