Please help! I don't know how to calculate?

Let #f(x) = 2^x#. Using the chain rule, determine an expression for the derivative of #[f(g(x))]#.

1 Answer
Geoff K.
Apr 26, 2018

If #g(x)# is unspecified, the expression for the derivative of #f[g(x)]# is

#d/dx f[g(x)] = ln 2*2^(g(x))*g'(x)#.

Explanation:

Using the generic function #g(x)#, we get

#f[g(x)] = 2^(g(x))#

Thus,

#d/dx f[g(x)] = d/dx 2^(g(x))#

The chain rule says: If #y# is a function of #u#, and #u# is a function of #x#, then #dy/dx = dy/(du) * (du)/dx#.

Basically, you treat #g(x)# as your variable when differentiating #f#, then use #x# when differentiating #g#.

#d/dx f[g(x)] = (df)/(dg) * (dg)/dx#

#color(white)(d/dx f[g(x)]) = d/(dg) 2^g * d/dx g(x)#

#color(white)(d/dx f[g(x)]) = (ln 2)(2^g) * g'(x)#

#color(white)(d/dx f[g(x)]) = (ln 2)(2^(g(x)))g'(x)#

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