# First Principles definition of Derivative: 1

#### math_fun

##### New member
I know that the derivative of 1 is obviously 0, but using the first principles definition of the derivative limit rules for a constant, it is 1. Where am I going wrong here? Thanks.

#### Subhotosh Khan

##### Super Moderator
Staff member
I know that the derivative of 1 is obviously 0, but using the first principles definition of the derivative limit rules for a constant, it is 1. Where am I going wrong here? Thanks.
We can correct your mistake only if you share your work - otherwise we would have to throw stones in dark!

#### pka

##### Elite Member
If $$f(x)=1$$ then $$\dfrac{f(x+h)-f(x)}{h}=\dfrac{1-1}{h}=\dfrac{0}{h}=~?$$

#### math_fun

##### New member
Isn't it that if f(x) = C, then lim x-> n = C? Then substitute C=1.

#### pka

##### Elite Member
Isn't it that if f(x) = C, then lim x-> n = C? Then substitute C=1.
Did you even read the question? It says the derivative of 1. That is $$f(x)=1$$.
In any case $$\dfrac{f(x+h)-f(x)}{h}=\dfrac{1-1}{h}=\dfrac{0}{h}=0$$
Thus $$\mathop {\lim }\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}=0$$

#### JeffM

##### Elite Member
I know that the derivative of 1 is obviously 0, but using the first principles definition of the derivative limit rules for a constant, it is 1. Where am I going wrong here? Thanks.
The first derivative rule for a constant is

$$\displaystyle f(x) = c \implies \lim_{h \rightarrow 0} \dfrac{f(x + h) - f(x)}{h} = \lim_{h \rightarrow 0} \dfrac{c- c}{h} =$$

$$\displaystyle \lim_{h \rightarrow 0} \dfrac{0}{h} =\lim_{h \rightarrow 0} 0 = 0.$$

You are mixing this up with a completely different rule.

$$\displaystyle f(x) = x \implies \lim_{h \rightarrow 0} \dfrac{f(x + h) - f(x)}{h} = \lim_{h \rightarrow 0} \dfrac{x + h- x}{h} =$$

$$\displaystyle \lim_{h \rightarrow 0} \dfrac{h}{h} =\lim_{h \rightarrow 0} 1 = 1.$$

#### HallsofIvy

##### Elite Member
I know that the derivative of 1 is obviously 0, but using the first principles definition of the derivative limit rules for a constant, it is 1. Where am I going wrong here? Thanks.
We can tell where you are going wrong when you don't show what you did!
Isn't it that if f(x) = C, then lim x-> n = C?
The limit of WHAT "as z=x->n? And what is "n"? Where did that come from? It is true that $$\displaystyle \lim_{x\to x_0} f(x)= f(x_0)= C$$ where $$\displaystyle x_0$$ is any value of C. That says that f(x)= C is continuous. It says nothing about the derivative!
Then substitute C=1.

#### Jomo

##### Elite Member
I am sure that you know that 1 is a constant. How can you say that the derivative of 1 is obviously 0 (WHY?) then say that the derivative of a constant is 1. Clearly you are wrong as you yourself found a counter example