Total derivative of $f(x_1, x_2, x_3) = x_1^{2x_2x_3}$

Question

Let $f: \{x \in \mathbb{R^3} \vert x_1 >0 \} \to \mathbb{R}$ and $f(x_1, x_2, x_3) = x_1^{2x_2x_3}$ find the total derivative of $f.$

The definition says that the total derivative can be expressed as a linear map the following way $$f(a+h) -f(a) = Df_a(h) + |h|\varepsilon(h)$$

However I’m not sure how to use this, the function given doesn’t seem to be linear in any sense? What’s the correct definition to use here?

Answer 1

Let $D=\{x_1>0\},$ viewed as a domain in $\mathbb R^3.$ Every $C^1$ function $f:D\to \mathbb R$ has a total derivative everywhere. And its total derivative is given by

$$Df(a,b,c)(v) = (\nabla f)(a,b,c)\cdot v.$$

So there's not much to do in our problem. Our function is $f(x_1,x_2,x_3) = \exp (2x_2x_3 \ln x_1).$ This is a composition of $C^1$ maps, hence is $C^1.$ So now just compute $\nabla f (a,b,c).$