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1. differentiating forms of a^x
hey guys I'm having trouble understanding how to differentiate a constant to the power of a variable, for example

y=5^x
I know
dy/dx = 5^x ln(5)

and this is the same for any integer, but I don't understand the mechanism and get stuck when it's something of the form

5x^x or 5^x^2
2. Re: differentiating forms of a^x
see I don't get how this follows :

y = x^5
ln Y = X ln (5)

differntiate both sides,

= 1/y = (x/5) + ln(5)

and then i'm confused.
3. Re: differentiating forms of a^x
(Original post by kingkongjaffa)
hey guys I'm having trouble understanding how to differentiate a constant to the power of a variable, for example

y=5^x
I know
dy/dx = 5^x ln(5)

and this is the same for any integer, but I don't understand the mechanism and get stuck when it's something of the form

5x^x or 5^x^2
Take logs and then differentiate.
4. Re: differentiating forms of a^x
I meant 5^x not x^5 which of course is 5x^4...
5. Re: differentiating forms of a^x
(Original post by steve2005)
Take logs and then differentiate.
can you show me I get stuck after taking logs?
6. Re: differentiating forms of a^x
Are you taking logs and then using implicit differentiation?

and so on?
7. Re: differentiating forms of a^x
(Original post by Mr M)
Are you taking logs and then using implicit differentiation?

and so on?
yeah i get stuck there the logs isn't a problem
8. Re: differentiating forms of a^x
(Original post by kingkongjaffa)
can you show me I get stuck after taking logs?
Here's a quicker way to do it.
1)Note that
2)Remember that
3) Letting yields
Last edited by Blutooth; 13-04-2012 at 12:22.
9. Re: differentiating forms of a^x
(Original post by kingkongjaffa)
can you show me I get stuck after taking logs?
As a general rule:

y = a^x

ln(y) = ln(a^x)

ln(y) = x ln(a)

Now differentiating - the differential of ln(x) is 1/x, so the differential of ln(y) is going to be 1/y times dy/dx:

1/y * dy/dx = ln(a) <- differentiated the RHS by treating ln(a) as a constant, just as 3x goes to 3 when you diff.

dy/dx = y ln(a)

but y = a^x

so dy/dx = a^x ln(a)
10. Re: differentiating forms of a^x
(Original post by kingkongjaffa)
yeah i get stuck there the logs isn't a problem

You know what y = ...?
11. Re: differentiating forms of a^x
ok i realised what i was doing wrong and understand how to do it in the form a^x without looking at the rule....
12. Re: differentiating forms of a^x
y=5^x
lny=ln(5^x)
lny=xln5

differentiate both sides, remembering ln5 is constant
13. Re: differentiating forms of a^x
How will one apply chain rule when its related to a fraction in a square root?