Chain rule in Calc = Chain in Log?

[PrudensOptimus]

I know in Logarithms loga b * logc d = loga d * logc b

and

loga b * logb c = loga c.

Chain Rule.

Now I read Calculus, I found out about the Chain rule, are they the same?? Looks like it. But because of my poor English reading, I couldn't understand the text. Can some one explain what Chain rule is?

[grady]

They are not related. if you have several functions as arguments to other functions like f( g( h(x) ) ), then the derivative of this is f'( g( h( x ) ) ) * g'( h( x ) ) * h'( x ) do you see the pattern? So for f(x) = 1/x and g(x) = ln(x) and h(x) = x2, f( g( h (x ) ) ) = 1/ln(x2) and the derivative would be -1/(ln(x2)2) * 1/(x2)*2x

[Dx]

great website (http://archives.math.utk.edu/visual.calculus/2/chain_rule.2/)

chain rule
Dx [;)]

[PrudensOptimus]

So what is the derivative of n^x, suppose n is a real number, and x is an unknown. And power rule does not apply to this situation because x is not a real number.

[KLscilevothma]

let f(x) = nx
ln f(x) = x ln n (take ln on both sides)
f '(x)/f(x) = ln n (take the first derivative on both sides)
f '(x) = f(x)*ln n = nxln n

PS
1) ln is natural log (base e), only natural log can be used in differentiation.

2) d/dx ln f(x) = f '(x)/f(x)