Chain rule in Calc = Chain in Log?

[PrudensOptimus]

I know in Logarithms log_a b * log_c d = log_a d * log_c b

and

log_a b * log_b c = log_a 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]

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) = n^x
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 = n^xln 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)