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# differentiating forms of a^x Watch

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

and so on?
7. (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. (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
9. (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. (Original post by kingkongjaffa)
yeah i get stuck there the logs isn't a problem

You know what y = ...?
11. 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. y=5^x
lny=ln(5^x)
lny=xln5

differentiate both sides, remembering ln5 is constant
13. How will one apply chain rule when its related to a fraction in a square root?

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Updated: April 13, 2012
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