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

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

1 Answer
Geoff K.
Apr 26, 2018

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