# Reference for proof of theorem on directional derivative

I require a reference for the following theorem, which is stated without proof in Matrix Groups for Undergraduates by Kristopher Tapp. In what follows, let $$p \in U \subseteq \mathbb R^m$$, $$v \in \mathbb R^m$$, and $$f: U \to \mathbb R^n$$. The directional derivative of $$f$$ in the direction $$v$$ at $$p$$ is defined in the usual way as:

$$df_p(v) = \lim_{t \to 0} \frac{f(p+tv)-f(p)}{t}$$

if this limit exists and is finite.

The theorem, which is stated after the above definition, is as follows:

If $$f$$ is $$C^1$$ on $$U$$, then for all $$p \in U$$,

(1) $$v \mapsto df_p(v)$$ is a linear function from $$\mathbb R^m$$ to $$\mathbb R^n$$.

(2) $$f(q) \approx f(p) + df_p(q-p)$$ is a good approximation of $$f$$ near $$p$$ in the following sense: for any infinite sequence $$\left\{ q_1, q_2,...\right\}$$ of points in $$\mathbb R^m$$ converging to $$p$$, $$\lim_{t \to \infty} \frac{f(q_t)-f(p)-df_p(q_t-p)}{|q_t-p|} = 0.$$

Immediately before stating this theorem the author asserts 'The following is proven in any real analysis textbook.' I know what he really means is 'any advanced calculus / vector analysis' textbook (rather than real analysis), and I have checked several but am so far unsuccessful in finding the result at all, let alone the proof. If anyone could point me to a reference, either online or textbook, where it is proved I'd be very grateful.

• Do you know the little-o notation ? Then by definition of the derivative $f(p+tv) = f(p)+t d f_p(v)+o(t)$, $df_p(tv) = t d_pf(v)$ and by continuity of the derivative $df_{p+tv}(u) = df_p(u) +o(1)$. Thus $$f(p)+t d_p f(v+u)+o(t)= f(p+tv+tu) = (f(p)+td_pf(v)+o(t))+t(d_pf(u)+o(1))+o(t)$$ which proves the linearity. – reuns Apr 26 at 13:44

Perhaps you are having trouble finding it because exposition in calculus/analysis books is usually in somewhat different order. One usually first defines $$f$$ to be differentiable at $$p$$ if there is a linear map $$df_p$$ such that the approximation $$f(q)\approx f(p)+df_p(q-p)$$ is good. Then one shows that 1) if $$f$$ is $$C^1$$ at $$p$$ (in Tapp's terminology, meaning continuous partials) then it is differentiaible at $$p$$; and 2) if $$f$$ is differentiable at $$p$$ then $$df_p(v)$$ is the directional derivative of $$f$$ in the direction of $$v$$. Combining these you do obtain the theorem in your question.