Question

What is d/dx(10^x) ?

Answer 1

Okay... you know what is
\[\Large \frac{d}{dx}e^x=\color{red}?\]

Answer 2

yep! - e^x itself.

Answer 3

That is correct :)
Now let's play with this expression a bit...
\[\Large 10^x =[ e^{\ln 10}]^x\]
Is this not so? :D

Answer 4

Well, anyway, as per your laws of exponents...
\[\Large 10^x = [e^{\ln 10}]^x=e^{(\ln10)x}\]
So you may now use the chain rule... not forgetting that ln(10) is JUST A CONSTANT.

\[\Large \frac{d}{dx}[e^{(\ln10)x}]=\color{red}?\]

Answer 5

the general rule is:\[\frac d{dx}[a^x]=a^x~ln(a)\]

when a=e, that simplifies to the e^x "rule"

Answer 6

^ Well you're no fun...
HAHA Just kidding
Peace ^_^

Answer 7

\[\Large \frac{d}{dx}[e^{(\ln10)x}]=e^{x\ln 10}\cdot \frac{d}{dx}(x\cdot \ln10)\]
\[\Large = e^{\ln 10^x}\cdot \ln 10\]
\[\Large = 10^x\cdot \ln 10\]
Just as it should be :)
Feel free to replace 10 with a XD

Answer 8

:) i asked my teacher why there is a constant rule AND and a power rule:

isnt kx^0 = k*0x^{-1} = 0?

but since there is a "what if x=0" undefinablility .... they tweaked it

Answer 9

Man, You're good, but to be honest, I didn't get much of what you said :'( - @terenzreignz

Answer 10

Oh, I rushed it. Just for you to appreciate the awesomeness of simplicity :D
I can go into more detail if you so desire :D

Answer 11

And, @amistre64 , Thanks a lot for the help, but i want an explanation on how exactly does it happen!

Answer 12

Here, let's go for the general case.
\[\Large \frac{d}{dx}a^x\]

Answer 13

Okay..

Answer 14

we only know the more 'traditional' derivatives plus the fact that\[\Large \frac{d}{dx}e^x = e^x\]
aye? :)

Answer 15

Aye aye :P

Answer 16

Okay. Then let's take a look at \(\large a^x\)
For starters,
\[\Large a = e^{\ln a}\]
right?

Answer 17

That part understood?

Answer 18

Working on it ! :P

Answer 19

I mean, it's part of the properties of logarithms...
\[\Huge b^{\log_bx}=x\\\Huge e^{\ln \ x}=x\]

Answer 20

Oohh kaay!
So we're just supposed to mug up the property, no ? :P
And, The rest is fine!

Thank You! :)

Answer 21

Are you sure you get it now? Well, the rest is there, just not in so much detail....