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#2 2005-12-27 12:24:08
- mathsyperson
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Re: derivative of an exponential function
You need the product rule: f'(uv) = f(u)f'(v) + f'(u)f(v)
So, (x^5 * 3^x)' = 3^x(x^5)' + x^5(3^x)'
(x^5)' = 5*x^4, using the basic differentiation of powers rule.
3^x is slightly different, because x is the exponent, so this time the derivative is ln 3* 3^x.
So, your overall derivative is 5 * x^4 * 3^x + x^5 * ln 3 * 3^x
Factorise to make it neater: x^4*3^x(5+x*ln 3)
Why did the vector cross the road?
It wanted to be normal.
#4 2005-12-27 17:31:22
- krassi_holmz
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Re: derivative of an exponential function
x^4*3^x(5+x*ln 3)=x^4*3^x*x(5/x+ln3)=x^5*3^x*(5/x+ln3)
IPBLE: Increasing Performance By Lowering Expectations.
#7 2005-12-27 18:30:47
- mikau
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Re: derivative of an exponential function
I'm not sure I've done differentiation of the form u^x yet. I think I know how to do it but I'm not sure so I'll let someone who knows tell you.
Last edited by mikau (2005-12-27 18:35:14)
A logarithm is just a misspelled algorithm.
#8 2005-12-27 18:43:12
- Flowers4Carlos
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Re: derivative of an exponential function
it's not that difficult really. just apply log to both sides and differentiate.
y = (cos x )^0.7x
lny = ln(cos x )^0.7x
lny = .7xln(cox)
give it a try! oh and happy holidays everyone!
#9 2005-12-28 01:12:18
- Ricky
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Re: derivative of an exponential function
d/dx (a^x) = a^x * ln(a)
You can use this, and the chain rule as well. Normally, solving an implicit differentiation (where you don't have y by itself alone on one side of the equation) can lead to some complications later, but in this case, the derivative of ln(y) is 1 / y, so it works out pretty well.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
#12 2005-12-29 12:40:52
- krassi_holmz
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Re: derivative of an exponential function
hristo, where are you from?
IPBLE: Increasing Performance By Lowering Expectations.
#14 2005-12-30 04:19:00
- krassi_holmz
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Re: derivative of an exponential function
Yes. And i'm asking because his name is like someone from East Europe.
IPBLE: Increasing Performance By Lowering Expectations.