Question

Derivative Q!

Answer 1

Ooo you're starting the fun stuff! :D yayy

Answer 2

Why is e^(1/x) a composite function?

Answer 3

Ha :D

Answer 4

It's pretty fun, I admit

Answer 5

WIsh I had a better calculus teacher tho :/ I don't thin she likes me

Answer 6

Ya it's the best! @rayman

Answer 7

\[\large\rm f(x)=e^{x}\qquad\qquad g(x)=\frac1x\]Your composition is \(\large\rm f(g(x))\)

Answer 8

Soooo, chain rule, ya?

Answer 9

Remember your rule for differentiating an exponential?
Hint: Not power rule! :O

Answer 10

Why is this a composite function though?
Like, why couldn't I bring down the 1/x and treat it like a product rule?

Answer 11

Because it's a composition :O
That's a weird question.
I guess you need to get comfortable identifying the difference between a composition and a product.

Examples of Compositions:
\(\large\rm sin\left(ln x\right)\)
\(\large\rm e^{x^2+3}\)

Examples of Products:
\(\large\rm sin(x)ln x\)
\(\large\rm x^2 e^x\)

Answer 12

I'm not exactly sure what you mean by "bring down the 1/x".
How would you pull that off? :d
Show me your magic.

Answer 13

By "bring down the 1/x" I meant the first answer on KA :P
lemme think about this composition stuff... thinking... hmm

Answer 14

`Like, why couldn't I bring down the 1/x and treat it like a product rule?`
I think you meant to say power rule
https://www.mathsisfun.com/calculus/derivatives-rules.html
but that only works if the exponent is a constant (not a variable expression)

Answer 15

^Yeah xd
I meant the power rule

Answer 16

Ya that only works when the exponent is constant,\[\large\rm \frac{d}{dx}x^c=c x^{c-1}\]Does not work for something like this:\[\large\rm \frac{d}{dx}x^x\ne x\cdot x^{x-1}\]where the exponent is varying.

Answer 17

What's your "fancy" rule called?

Answer 18

You are working with an exponential function here, not a power function. Therefore, please stick to the derivative formulas applying to expo functions, remembering that we also have to apply the chain rule (because the exponent of your 'e' is itself a function).

Answer 19

You've learned derivative of e^x at this point, yes? :)

Answer 20

\[f(x)=e ^{\frac{ 1 }{ x }}\]

Answer 21

Yeah zep xD

Answer 22

It gives you the same thing back that you started with.\[\large\rm \frac{d}{dx}e^x=e^x\]Since our exponent is `more than just x`, we have to chain,\[\large\rm \frac{d}{dx}e^{\frac1x}=e^{\frac1x}\frac{d}{dx}\frac1x\]Multiplying by the derivative of the "inner" function.

Answer 23

Great. if you differentiate y=e^x, you get e^x as your derivative.

But if your exponent on that 'e' is itself a function, you must ALSO diffrerentiate that function. zepdrix has shown this clearly and correctly (above).

Answer 24

Alrighty, let's get down to it :D

Answer 25

Great. Your move. Evaluate:\[\frac{ d }{ dx}e^u\]

Answer 26

where u represents a separate function of x, namely, u=1/x.

Answer 27

Reminder: Find the derivative of e^u with respect to u first, and then multiply that result by the derivative of u (which in your problem is the derivative of (1/x) ).

Answer 28

If you want to look up explanations of this material, search for "Differentiation of Composite Functions."

Answer 29

"Like, why couldn't I bring down the 1/x and treat it like a product rule?"

At the risk of repeating myself, you are dealing with an EXPONENTIAL FUNCTION, which is a whole different animal from a POWER FUNCTION. That's why you cannot "bring down that (1/x) and multiply e^(1/x) by it.

In the most basic case, the BASE of the POWER FUNCTION is a variable, such as x, or a function. In contrast, the BASE of the EXPONENTIAL FUNCTION is a constant, and that constant is e. Reminder: e is approx. equal to 2.7182818....