Show that $d/dx (a^x) = a^xln a$.
up vote
9
down vote
favorite
Show that
$$
frac{d}{dx} a^x = a^x ln a.
$$
How would I do a proof for this. I can't seem to get it to work anyway I try.
I know that
$$
frac{d}{dx} e^x = e^x.
$$
Does that help me here?
calculus
add a comment |
up vote
9
down vote
favorite
Show that
$$
frac{d}{dx} a^x = a^x ln a.
$$
How would I do a proof for this. I can't seem to get it to work anyway I try.
I know that
$$
frac{d}{dx} e^x = e^x.
$$
Does that help me here?
calculus
add a comment |
up vote
9
down vote
favorite
up vote
9
down vote
favorite
Show that
$$
frac{d}{dx} a^x = a^x ln a.
$$
How would I do a proof for this. I can't seem to get it to work anyway I try.
I know that
$$
frac{d}{dx} e^x = e^x.
$$
Does that help me here?
calculus
Show that
$$
frac{d}{dx} a^x = a^x ln a.
$$
How would I do a proof for this. I can't seem to get it to work anyway I try.
I know that
$$
frac{d}{dx} e^x = e^x.
$$
Does that help me here?
calculus
calculus
edited May 22 '13 at 3:48
kahen
13.2k32554
13.2k32554
asked May 22 '13 at 3:25
1ftw1
3602615
3602615
add a comment |
add a comment |
7 Answers
7
active
oldest
votes
up vote
16
down vote
Hint: $a^x=e^{ln a^x}=e^{xln a}$.
add a comment |
up vote
5
down vote
Let $f : mathbb{R} to mathbb{R}$ be given by $f(x)=a^x$ and consider the $ln$ function. We can take the composition so that we have:
$$(lncirc f)(x)=ln (a^x)=xln a$$
Now, if we take the derivative, on the left hand side we use the chain rule and on the right hand side we differentiate as usual so that we have:
$$frac{f'(x)}{f(x)}=ln a$$
Now solving for $f'(x)$ gives $f'(x) = f(x) ln a$ so that $f'(x) = a^x ln a$. This useful technique can be used to take derivatives of other functions: we compose the original function with the inverse and then differentiate on both sides and use the same idea we've used here, this technique can simplify many derivatives and save a lot of time in some situations.
add a comment |
up vote
3
down vote
Hint: Write $y = a^x$ or equivalently $ln y = x ln a$ and use implicit differentiation.
add a comment |
up vote
1
down vote
, the last term, , can be explained by L'Hopital's rule, just taking the limits of the numerator and denominator one of each time.
add a comment |
up vote
0
down vote
Here are the steps
$$
frac{d}{dx} left[a^xright] = frac{d}{dx} left[e^{ln a^x}right]= e^{ln a^x} frac{d}{dx} left[ln a^xright]
$$
$$
= a^x frac{d}{dx} left[xln aright] = a^xleft(ln aright)frac{d}{dx} left[xright]= a^xln a
$$
add a comment |
up vote
0
down vote
$ frac{d}{dx} a^x=$
$=d/dx$ $e^{ln (a^x)}=$
$=d/dx e^{xln(a)}$ just as you said.
Now it's time for chain rule.
$f(x)=e^x$
$g(x)=x*ln(a)$
$a^x= f(g(x))$
last comment before I get solving
$a=e^{ln(a)}$
now let's get it done
$d/dx e^{x ln(a)}=$
$=e^{x*ln(a)} d/dx (x ln(a))=$ (by chain rule)
$=e^{x ln(a)} * ln(a)$
and that is the solution. Wait it doesnt look correct --> we should do some algebra!
Show for yourself that
$e^{x*ln(a)}=( e^{ln(a)} )^x$ but as we said that $a = e^{ln(a)}$ so $( e^{ln(a)} )^x$ actually equals $a^x$ and of course $e^{x*ln(a)}$ equals a^x
With what we just said you can see that $e^{x * ln(a)} ln(a)$ equals $a^x ln(a)$ or
$$d/dx a^x = a^x ln(a)$$
1
Use TeX while typing math symbols
– supremum
Jan 5 '15 at 17:59
For help, see this FAQ entry about $LaTeX$ on Math.SE.
– Ruslan
Jan 5 '15 at 18:43
add a comment |
up vote
-1
down vote
Let $y = a^x$
Then taking log on both side to the base $e$
We have;
$ ln y = ln a^x$
$ ln y = xcdot ln a$
Taking derivative with respect to $x$;
$frac d{dx} ln y = frac d{dx} xcdot ln a$
(I have applied chain rule here>>)
$ frac 1y frac{dy}{dx} = ln a + 0$
$ frac {dy}{dx} = ycdot ln a$
We know $y= a^x$
So, $frac {dy}{dx} = a^x ln a$
Hope it was easier than other methodologies used to derive it!
The question is five years old and your answer adds nothing new to the already existing answers.
– José Carlos Santos
Nov 21 at 9:39
add a comment |
7 Answers
7
active
oldest
votes
7 Answers
7
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
16
down vote
Hint: $a^x=e^{ln a^x}=e^{xln a}$.
add a comment |
up vote
16
down vote
Hint: $a^x=e^{ln a^x}=e^{xln a}$.
add a comment |
up vote
16
down vote
up vote
16
down vote
Hint: $a^x=e^{ln a^x}=e^{xln a}$.
Hint: $a^x=e^{ln a^x}=e^{xln a}$.
answered May 22 '13 at 3:38
A. Chu
6,97783183
6,97783183
add a comment |
add a comment |
up vote
5
down vote
Let $f : mathbb{R} to mathbb{R}$ be given by $f(x)=a^x$ and consider the $ln$ function. We can take the composition so that we have:
$$(lncirc f)(x)=ln (a^x)=xln a$$
Now, if we take the derivative, on the left hand side we use the chain rule and on the right hand side we differentiate as usual so that we have:
$$frac{f'(x)}{f(x)}=ln a$$
Now solving for $f'(x)$ gives $f'(x) = f(x) ln a$ so that $f'(x) = a^x ln a$. This useful technique can be used to take derivatives of other functions: we compose the original function with the inverse and then differentiate on both sides and use the same idea we've used here, this technique can simplify many derivatives and save a lot of time in some situations.
add a comment |
up vote
5
down vote
Let $f : mathbb{R} to mathbb{R}$ be given by $f(x)=a^x$ and consider the $ln$ function. We can take the composition so that we have:
$$(lncirc f)(x)=ln (a^x)=xln a$$
Now, if we take the derivative, on the left hand side we use the chain rule and on the right hand side we differentiate as usual so that we have:
$$frac{f'(x)}{f(x)}=ln a$$
Now solving for $f'(x)$ gives $f'(x) = f(x) ln a$ so that $f'(x) = a^x ln a$. This useful technique can be used to take derivatives of other functions: we compose the original function with the inverse and then differentiate on both sides and use the same idea we've used here, this technique can simplify many derivatives and save a lot of time in some situations.
add a comment |
up vote
5
down vote
up vote
5
down vote
Let $f : mathbb{R} to mathbb{R}$ be given by $f(x)=a^x$ and consider the $ln$ function. We can take the composition so that we have:
$$(lncirc f)(x)=ln (a^x)=xln a$$
Now, if we take the derivative, on the left hand side we use the chain rule and on the right hand side we differentiate as usual so that we have:
$$frac{f'(x)}{f(x)}=ln a$$
Now solving for $f'(x)$ gives $f'(x) = f(x) ln a$ so that $f'(x) = a^x ln a$. This useful technique can be used to take derivatives of other functions: we compose the original function with the inverse and then differentiate on both sides and use the same idea we've used here, this technique can simplify many derivatives and save a lot of time in some situations.
Let $f : mathbb{R} to mathbb{R}$ be given by $f(x)=a^x$ and consider the $ln$ function. We can take the composition so that we have:
$$(lncirc f)(x)=ln (a^x)=xln a$$
Now, if we take the derivative, on the left hand side we use the chain rule and on the right hand side we differentiate as usual so that we have:
$$frac{f'(x)}{f(x)}=ln a$$
Now solving for $f'(x)$ gives $f'(x) = f(x) ln a$ so that $f'(x) = a^x ln a$. This useful technique can be used to take derivatives of other functions: we compose the original function with the inverse and then differentiate on both sides and use the same idea we've used here, this technique can simplify many derivatives and save a lot of time in some situations.
edited Jan 5 '15 at 23:28
answered May 22 '13 at 3:58
user1620696
11.3k340113
11.3k340113
add a comment |
add a comment |
up vote
3
down vote
Hint: Write $y = a^x$ or equivalently $ln y = x ln a$ and use implicit differentiation.
add a comment |
up vote
3
down vote
Hint: Write $y = a^x$ or equivalently $ln y = x ln a$ and use implicit differentiation.
add a comment |
up vote
3
down vote
up vote
3
down vote
Hint: Write $y = a^x$ or equivalently $ln y = x ln a$ and use implicit differentiation.
Hint: Write $y = a^x$ or equivalently $ln y = x ln a$ and use implicit differentiation.
answered May 22 '13 at 3:28
response
4,69121220
4,69121220
add a comment |
add a comment |
up vote
1
down vote
, the last term, , can be explained by L'Hopital's rule, just taking the limits of the numerator and denominator one of each time.
add a comment |
up vote
1
down vote
, the last term, , can be explained by L'Hopital's rule, just taking the limits of the numerator and denominator one of each time.
add a comment |
up vote
1
down vote
up vote
1
down vote
, the last term, , can be explained by L'Hopital's rule, just taking the limits of the numerator and denominator one of each time.
, the last term, , can be explained by L'Hopital's rule, just taking the limits of the numerator and denominator one of each time.
answered Feb 24 '14 at 0:22
user131054
111
111
add a comment |
add a comment |
up vote
0
down vote
Here are the steps
$$
frac{d}{dx} left[a^xright] = frac{d}{dx} left[e^{ln a^x}right]= e^{ln a^x} frac{d}{dx} left[ln a^xright]
$$
$$
= a^x frac{d}{dx} left[xln aright] = a^xleft(ln aright)frac{d}{dx} left[xright]= a^xln a
$$
add a comment |
up vote
0
down vote
Here are the steps
$$
frac{d}{dx} left[a^xright] = frac{d}{dx} left[e^{ln a^x}right]= e^{ln a^x} frac{d}{dx} left[ln a^xright]
$$
$$
= a^x frac{d}{dx} left[xln aright] = a^xleft(ln aright)frac{d}{dx} left[xright]= a^xln a
$$
add a comment |
up vote
0
down vote
up vote
0
down vote
Here are the steps
$$
frac{d}{dx} left[a^xright] = frac{d}{dx} left[e^{ln a^x}right]= e^{ln a^x} frac{d}{dx} left[ln a^xright]
$$
$$
= a^x frac{d}{dx} left[xln aright] = a^xleft(ln aright)frac{d}{dx} left[xright]= a^xln a
$$
Here are the steps
$$
frac{d}{dx} left[a^xright] = frac{d}{dx} left[e^{ln a^x}right]= e^{ln a^x} frac{d}{dx} left[ln a^xright]
$$
$$
= a^x frac{d}{dx} left[xln aright] = a^xleft(ln aright)frac{d}{dx} left[xright]= a^xln a
$$
answered Jan 5 '15 at 18:06
k170
7,44131540
7,44131540
add a comment |
add a comment |
up vote
0
down vote
$ frac{d}{dx} a^x=$
$=d/dx$ $e^{ln (a^x)}=$
$=d/dx e^{xln(a)}$ just as you said.
Now it's time for chain rule.
$f(x)=e^x$
$g(x)=x*ln(a)$
$a^x= f(g(x))$
last comment before I get solving
$a=e^{ln(a)}$
now let's get it done
$d/dx e^{x ln(a)}=$
$=e^{x*ln(a)} d/dx (x ln(a))=$ (by chain rule)
$=e^{x ln(a)} * ln(a)$
and that is the solution. Wait it doesnt look correct --> we should do some algebra!
Show for yourself that
$e^{x*ln(a)}=( e^{ln(a)} )^x$ but as we said that $a = e^{ln(a)}$ so $( e^{ln(a)} )^x$ actually equals $a^x$ and of course $e^{x*ln(a)}$ equals a^x
With what we just said you can see that $e^{x * ln(a)} ln(a)$ equals $a^x ln(a)$ or
$$d/dx a^x = a^x ln(a)$$
1
Use TeX while typing math symbols
– supremum
Jan 5 '15 at 17:59
For help, see this FAQ entry about $LaTeX$ on Math.SE.
– Ruslan
Jan 5 '15 at 18:43
add a comment |
up vote
0
down vote
$ frac{d}{dx} a^x=$
$=d/dx$ $e^{ln (a^x)}=$
$=d/dx e^{xln(a)}$ just as you said.
Now it's time for chain rule.
$f(x)=e^x$
$g(x)=x*ln(a)$
$a^x= f(g(x))$
last comment before I get solving
$a=e^{ln(a)}$
now let's get it done
$d/dx e^{x ln(a)}=$
$=e^{x*ln(a)} d/dx (x ln(a))=$ (by chain rule)
$=e^{x ln(a)} * ln(a)$
and that is the solution. Wait it doesnt look correct --> we should do some algebra!
Show for yourself that
$e^{x*ln(a)}=( e^{ln(a)} )^x$ but as we said that $a = e^{ln(a)}$ so $( e^{ln(a)} )^x$ actually equals $a^x$ and of course $e^{x*ln(a)}$ equals a^x
With what we just said you can see that $e^{x * ln(a)} ln(a)$ equals $a^x ln(a)$ or
$$d/dx a^x = a^x ln(a)$$
1
Use TeX while typing math symbols
– supremum
Jan 5 '15 at 17:59
For help, see this FAQ entry about $LaTeX$ on Math.SE.
– Ruslan
Jan 5 '15 at 18:43
add a comment |
up vote
0
down vote
up vote
0
down vote
$ frac{d}{dx} a^x=$
$=d/dx$ $e^{ln (a^x)}=$
$=d/dx e^{xln(a)}$ just as you said.
Now it's time for chain rule.
$f(x)=e^x$
$g(x)=x*ln(a)$
$a^x= f(g(x))$
last comment before I get solving
$a=e^{ln(a)}$
now let's get it done
$d/dx e^{x ln(a)}=$
$=e^{x*ln(a)} d/dx (x ln(a))=$ (by chain rule)
$=e^{x ln(a)} * ln(a)$
and that is the solution. Wait it doesnt look correct --> we should do some algebra!
Show for yourself that
$e^{x*ln(a)}=( e^{ln(a)} )^x$ but as we said that $a = e^{ln(a)}$ so $( e^{ln(a)} )^x$ actually equals $a^x$ and of course $e^{x*ln(a)}$ equals a^x
With what we just said you can see that $e^{x * ln(a)} ln(a)$ equals $a^x ln(a)$ or
$$d/dx a^x = a^x ln(a)$$
$ frac{d}{dx} a^x=$
$=d/dx$ $e^{ln (a^x)}=$
$=d/dx e^{xln(a)}$ just as you said.
Now it's time for chain rule.
$f(x)=e^x$
$g(x)=x*ln(a)$
$a^x= f(g(x))$
last comment before I get solving
$a=e^{ln(a)}$
now let's get it done
$d/dx e^{x ln(a)}=$
$=e^{x*ln(a)} d/dx (x ln(a))=$ (by chain rule)
$=e^{x ln(a)} * ln(a)$
and that is the solution. Wait it doesnt look correct --> we should do some algebra!
Show for yourself that
$e^{x*ln(a)}=( e^{ln(a)} )^x$ but as we said that $a = e^{ln(a)}$ so $( e^{ln(a)} )^x$ actually equals $a^x$ and of course $e^{x*ln(a)}$ equals a^x
With what we just said you can see that $e^{x * ln(a)} ln(a)$ equals $a^x ln(a)$ or
$$d/dx a^x = a^x ln(a)$$
edited Jan 9 '15 at 20:35
Community♦
1
1
answered Jan 5 '15 at 17:29
grekiki
1
1
1
Use TeX while typing math symbols
– supremum
Jan 5 '15 at 17:59
For help, see this FAQ entry about $LaTeX$ on Math.SE.
– Ruslan
Jan 5 '15 at 18:43
add a comment |
1
Use TeX while typing math symbols
– supremum
Jan 5 '15 at 17:59
For help, see this FAQ entry about $LaTeX$ on Math.SE.
– Ruslan
Jan 5 '15 at 18:43
1
1
Use TeX while typing math symbols
– supremum
Jan 5 '15 at 17:59
Use TeX while typing math symbols
– supremum
Jan 5 '15 at 17:59
For help, see this FAQ entry about $LaTeX$ on Math.SE.
– Ruslan
Jan 5 '15 at 18:43
For help, see this FAQ entry about $LaTeX$ on Math.SE.
– Ruslan
Jan 5 '15 at 18:43
add a comment |
up vote
-1
down vote
Let $y = a^x$
Then taking log on both side to the base $e$
We have;
$ ln y = ln a^x$
$ ln y = xcdot ln a$
Taking derivative with respect to $x$;
$frac d{dx} ln y = frac d{dx} xcdot ln a$
(I have applied chain rule here>>)
$ frac 1y frac{dy}{dx} = ln a + 0$
$ frac {dy}{dx} = ycdot ln a$
We know $y= a^x$
So, $frac {dy}{dx} = a^x ln a$
Hope it was easier than other methodologies used to derive it!
The question is five years old and your answer adds nothing new to the already existing answers.
– José Carlos Santos
Nov 21 at 9:39
add a comment |
up vote
-1
down vote
Let $y = a^x$
Then taking log on both side to the base $e$
We have;
$ ln y = ln a^x$
$ ln y = xcdot ln a$
Taking derivative with respect to $x$;
$frac d{dx} ln y = frac d{dx} xcdot ln a$
(I have applied chain rule here>>)
$ frac 1y frac{dy}{dx} = ln a + 0$
$ frac {dy}{dx} = ycdot ln a$
We know $y= a^x$
So, $frac {dy}{dx} = a^x ln a$
Hope it was easier than other methodologies used to derive it!
The question is five years old and your answer adds nothing new to the already existing answers.
– José Carlos Santos
Nov 21 at 9:39
add a comment |
up vote
-1
down vote
up vote
-1
down vote
Let $y = a^x$
Then taking log on both side to the base $e$
We have;
$ ln y = ln a^x$
$ ln y = xcdot ln a$
Taking derivative with respect to $x$;
$frac d{dx} ln y = frac d{dx} xcdot ln a$
(I have applied chain rule here>>)
$ frac 1y frac{dy}{dx} = ln a + 0$
$ frac {dy}{dx} = ycdot ln a$
We know $y= a^x$
So, $frac {dy}{dx} = a^x ln a$
Hope it was easier than other methodologies used to derive it!
Let $y = a^x$
Then taking log on both side to the base $e$
We have;
$ ln y = ln a^x$
$ ln y = xcdot ln a$
Taking derivative with respect to $x$;
$frac d{dx} ln y = frac d{dx} xcdot ln a$
(I have applied chain rule here>>)
$ frac 1y frac{dy}{dx} = ln a + 0$
$ frac {dy}{dx} = ycdot ln a$
We know $y= a^x$
So, $frac {dy}{dx} = a^x ln a$
Hope it was easier than other methodologies used to derive it!
edited Nov 21 at 9:46
idea
1,96231024
1,96231024
answered Nov 21 at 9:17
Vichitra Attri
11
11
The question is five years old and your answer adds nothing new to the already existing answers.
– José Carlos Santos
Nov 21 at 9:39
add a comment |
The question is five years old and your answer adds nothing new to the already existing answers.
– José Carlos Santos
Nov 21 at 9:39
The question is five years old and your answer adds nothing new to the already existing answers.
– José Carlos Santos
Nov 21 at 9:39
The question is five years old and your answer adds nothing new to the already existing answers.
– José Carlos Santos
Nov 21 at 9:39
add a comment |
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