# Directional derivatives - $t \mapsto f(x+tv)$

First I will write what is written in the book and then I will ask the question:

Suppose $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ is a real-valued function. Let $v$ and $x \in \mathbb{R}^3$ be fixet vectors and consider the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $t \mapsto f(x+tv)$. The set of points of the form $x+tv, t\in \mathbb{R}$ is the line $L$ through the point $x$ parallel to the vector $v$.

The function $t \mapsto f(x+tv)$ represents the function $f$ restricted to the line $L$. For example, is a bird flies along this line with velocity $v$ so that $x+tv$ is its position at time $t$, and if $f$ represents the temperature as a function of position, then $f(x+tv)$ is the temperature at time $t$. We may ask: How fast are the values of $f$ changing along the line $L$ at the point $x$ ? Because the rate of change of a function is given by a derivative, we could say that the answer to this question is the value of the derivative of this function of $t$ at $t=0$ (when $t=0$, $x+tv$ reduce to $x$). This would be the derivative of $f$ at the point $x$ in the direction of $L$, that is, of $v$.

What is written in bold is my issue. I don't understand the question very well and I can't translate the answer from the text correctly. Why is the answer to that question the value of the derivative of that function of $t$ at $t=0$. Why $t=0$ and not $t=1$?

Thanks :)

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## 2 Answers

I don't know if this will make things more clear for you; hopefully it will help a bit.

The derivative of a function $f: \mathbb{R} \to \mathbb{R}$ at a point $a$ is (as you probably already know): $$f'(a) = \lim_{x \to a}\frac{f(x) - f(a)}{ x- a} = \lim_{h \to 0}\frac{f(a + h) - f(a)}{h}.$$

So we look at the difference quotient while we approach the point in questions.

Now for a function $f: \mathbb{R}^3 \to \mathbb{R}$ we also have derivatives. But now there is a direction on the derivative. Consider the point $\bf{a} \in \mathbb{R}^3$ and the direction from that point given by a direction vector $\vec{v}$. Again we want to consider a difference quotient while we vary the $\bf{x}$ along the line with (unit) direction vector $\vec{v}$. That is we are considering the line $L$ given by teh vector equation: $\bf{a} + t\vec{v}$. So looking at the values of $f$ along that line we get the directional derivative: $$\lim_{t \to 0}\frac{f(\bf{a} + t\vec{v}) - f(\bf{a})}{t}.$$

We need $t$ to approach $0$ because we want the values of $f$ near the point $\bf{a}$.

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It is not at $t=0$ nor $t=1$, $t$ is the variable, and you take the limit $t\to 0$ to define the derivative.

When you write the derivative, $t$ doesn't appear anymore. It's written $\partial_{L}f(x)$. It's just like a real function: $f(x)$, when you define the derivative, you take the limit as $\varepsilon$ goes to $0$:

$$f'(x) = \lim_{\varepsilon \to 0} \frac{f(x+\varepsilon)-f(x)}{\varepsilon}$$

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