Question

use the definition of the derivative to find f'(x) ; f(x)=-5x

Answer 1

Do you know what is the definition of the derivative?

Answer 2

yes. lim f(x+deltax)-f(x) / delta x as delta x approaches 0

Answer 3

Excellent, can you try to plug the function into that formula, and solve?

Answer 4

uhh when i plugged it in i got 0. but the back of the book says -5

Answer 5

The answer is indeed -5

Answer 6

Okay, let's go with a fact to make this all ezpz :D\[\large \textbf{FACT:} \\ \large \frac{d}{dx}f(x)=\lim _{\Delta x \rightarrow0}\frac{f(x+\Delta x)-f(x)}{\Delta x}\\\large \frac{d}{dx}cf(x) = c\frac{d}{dx}f(x)=c*\lim _{\Delta x \rightarrow0}\frac{f(x+\Delta x)-f(x)}{\Delta x}, \textrm{Where c is a constant.}\]

Answer 7

Let's use this fact to solve our problem: \[\large \frac{d}{dx}f(x)=\frac{d}{dx}(-5x)\]Since -5 is just a constant, we can simply factor it out of the derivative: \[\large -5(\frac{d}{dx}x)\]

Answer 8

umm what's d?

Answer 9

\(\LARGE \frac{d}{dx}\) is a just a notation to say 'Derivative of'.

Answer 10

With respect to x.

Answer 11

So, \[\large -5(\frac{d}{dx}x)=-5*\lim _{\Delta x \rightarrow0}\frac{x+\Delta x-x}{\Delta x}\]Can you go on and solve it now?

Answer 12

you plug -5 into x right??

Answer 13

-5 is a constant and it's factored out of the derivative, so you only have to solve the limit now.

Answer 14

oh then it's -5

Answer 15

The result to the limit is simply 1, 1*-5 = -5

Answer 16

wait how did you factor it out of the derivative

Answer 17

Because a constant does not affect the limiting process.

Answer 18

ohhhhh

Answer 19

yep yep yep :)

Answer 20

how come when i plug -5x in the equation it ends up as 0??

Answer 21

Let's see:
\[\large \begin{align}
&\frac{d}{dx}(-5x)\\
&=\lim _{\Delta x \rightarrow0}\frac{-5(x+\Delta x)-(-5x)}{\Delta x}\\
&=\lim _{\Delta x \rightarrow0}\frac{-5x-5\Delta x+5x}{\Delta x}\\
&=\lim _{\Delta x \rightarrow0}\frac{-5\Delta x}{\Delta x}\\
&=\lim _{\Delta x \rightarrow0}-5\\
&\textbf{The limit is now just useless, take it off.}\\
&=-5
\end{align}\]

Answer 22

wait why did you put -5 in front of (x+delta x)

Answer 23

Because I put -5x all together in the limit, and replace the x in this function by x+ delta x.

Answer 24

wait what do you mean by that

Answer 25

The notation f(x+delta x) mean that you'll need to replace the x in the function by the value in the parentheses, x+delta x, therefore, if you have -5x, f(delta x+ x) would be -5(delta x+ x)

Answer 26

WTF I DID NOT KNOW THAT UGHHHHHH OKAY

Answer 27

OK THANK YOU!!!!

Answer 28

Alright, no problem :)