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Petrus
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How do i derivate e^e^x (I don't know how to type it on latex but here you can se what i mean e^e^x - Wolfram|Alpha Results basicly don't know how I shall think
\(\displaystyle e^{f(x)}*f'(x)\)ZaidAlyafey said:If you have something of the form \(\displaystyle e^{f(x)}\) what is the derivative ?
Petrus said:How do i derivate e^e^x (I don't know how to type it on latex but here you can se what i mean e^e^x - Wolfram|Alpha Results basicly don't know how I shall think
Petrus said:\(\displaystyle e^{f(x)}*f'(x)\)
$e^{e^x}*e^x$ZaidAlyafey said:Excellent , now put \(\displaystyle f(x)=e^x\) .
Hello!chisigma said:Some of most 'horrors' I have seen in my life are the so called 'towers of powers'... when I see an expression like $\displaystyle e^{e^{x}}$ I don't undestand if it is $\displaystyle e^{(e^{x})}$ or $\displaystyle (e^{e})^{x}$ (Headbang)...
Kind regards
$\chi$ $\sigma$
Petrus said:Hello!
I think this is hilirious because I got one problem like thathttps://www.physicsforums.com/attachments/681
anyone can give me advice on that aswell. I am suposed to derivate it once.
If u mean 2^x and derivate is ln(2)•2^xI like Serena said:So... how far do you get?
Do you know what the derivative of $2^y$ is?
Petrus said:If u mean 2^x and derivate is ln(2)•2^x
Im really confused on this because its a lot Power up to. Can you give me a adviceI like Serena said:Good!
So...?
Petrus said:Im really confused on this because its a lot Power up to. Can you give me a advice
I need first help how I shall think when i derivate $3^{x^2}$I like Serena said:Okay.
Let's start by picking $f(y)=2^y$ and $g(x)=3^{x^2}$.
Can you apply the chain rule?
Petrus said:I need first help how I shall think when i derivate $3^{x^2}$
$ln(3)3^y2x$I like Serena said:Pick $f(y)=3^y$ and $g(x)=x^2$ this time round.
Can you apply the chain rule to that?
Petrus said:$ln(3)3^y2x$
$ln(3)3^{x^2}2x$I like Serena said:Almost.
When substituting in $f'(g(x))\cdot g'(x)$, you're supposed to replace all occurrence of $y$ in $f'(y)$ by $g(x)$.
That is, you're supposed to replace $y$ in $\ln(3) 3^y$ by $x^2$.
Can you do that?
Petrus said:$ln(3)3^{x^2}2x$
(im replying to this cause i want to use it as a mall)I like Serena said:Okay.
Let's start by picking $f(y)=2^y$ and $g(x)=3^{x^2}$.
Can you apply the chain rule?
Petrus said:(im replying to this cause i want to use it as a mall)
I hope this trick work, I subsitate $x^2$ as c and $c'=2x$
$ln(2)2^yln(3)3^c*c'$
$ln(2)2^yln(3)3^{x^2}2x$
Hello,I like Serena said:Looks good... almost there...
There is an $y$ left that should still be replaced by the $g(x)$...
Petrus said:Hello,
Could you give me latex code for like 2^x^(y^2)
I can't replace that y with $3^{x^2}$ with latex I get error
$ln(2)2^{\displaystyle 3^{\displaystyle x^{\displaystyle2}}}ln(3)3^{\displaystyle x^2}2x$I like Serena said:You need curly braces {} to group symbols in latex.
So 2^x^(y^2) is 2^{x^{y^2}} which looks like $2^{x^{y^2}}$.
By adding a couple of \displaystyle directives we get $2^{\displaystyle x^{\displaystyle y^2}}$, which is more readable.
We need to be careful with these towers, because they look like the are on the verge of falling over. ;)
For safety we could keep them on the ground, like 2^(x^(y^2)).
An exponential function is a mathematical function in the form of f(x) = ab^x, where a is a constant and b is the base of the exponent. These functions grow or decay at a constant rate as x increases or decreases, making them useful for modeling growth and decay processes in various fields such as science, finance, and engineering.
To differentiate an exponential function, you can use the power rule which states that for any function f(x) = ax^n, the derivative is f'(x) = nax^(n-1). For an exponential function, this becomes f'(x) = a(ln b)b^x. This means that the derivative of an exponential function is proportional to the original function and its constant base.
Differentiating exponential functions is useful in many applications, such as finding the slope of a curve at a specific point, determining the growth or decay rate of a process, and solving optimization problems. It also allows us to analyze the behavior of exponential functions and make predictions about their future values.
Yes, an exponential function can be differentiated as many times as needed. Each time, the derivative will be proportional to the original function and its base. This means that the rate of change of an exponential function increases or decreases depending on its base, making it a powerful tool for modeling various real-world phenomena.
An exponential function is the inverse of a logarithmic function, and vice versa. While an exponential function has a variable in the exponent, a logarithmic function has a variable in the base. Exponential functions grow or decay at a constant rate, while logarithmic functions grow or decay at an increasing rate. Additionally, an exponential function maps inputs to outputs, while a logarithmic function maps outputs to inputs.