Question

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))]`.

Answer 1

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)`