Differential of scalar product composed with differentiable function

Question

I am looking to find the differential of function $$h(x)=<f(x),x>$$ where $f$ is a differentiable function $f:\mathbb R^n \to \mathbb R^n$ .

I tried the following:

$$Dh(x)k=<Df(x)k,x>+<f(x),k>$$

Is this correct? And if it isn't I would kindly ask you for a correct solution. Thanks for your help!

Answer 1

We have :

$$h(x+h)-h(x) = \langle f(x+h), x+h \rangle - \langle f(x), x \rangle = \langle f(x+h)-f(x), x \rangle + \langle f(x+h), h \rangle$$

Since the function $f$ is differentiable we have near $0$ :

$$ \langle f(x+h)-f(x), x \rangle + \langle f(x+h), h \rangle = \langle \mathrm{d}f(x)h, x \rangle + \langle \mathrm{d}f(x)h,h \rangle + \langle f(x), h \rangle + o( \| h \|) = \langle \mathrm{d}f(x)h, x \rangle + \langle f(x), h \rangle + o( \| h \|) $$

Hence we get :

$$\mathrm{d}h(x)(k)= \langle \mathrm{d}f(x)k, x \rangle + \langle f(x), k \rangle$$

Your answer is thus correct.