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i'm starting my calculus's journey and i have a question.

What does mean the exact value of a derivative Take an easy example we have a derivative of $f(x)=x^2$, that is $f'(x)=2x$.

Someone would say that for every $x$, $f'(x)$ is tanget line of $f(x)$ with respect to $x$ And this is true, graphically it looks like that, i don't have doubts here, it is just a graphical representation.

Next one would say that $f'(x)$ is how fast the function changes. But what does mean? for a linear function it describes how the function changes, say we have a linear function $g(x)$, if we change the argument $x$ by $h$, then $g(x)$ value changes by $h*g'(x)$. if we go by analogy, then if $f(x)$ changes by $h$ then $f(x)$ changes by $h*f'(x)$, but for every $x$ it would be only a approximation.

Let's compare $f(x)$ to car's traveled distance, then $f'(x)$ would be iinstantaneous speed of the car. Before derivatives car's speed was saying me that "How far i would travel by some amount of time, if speed doesn't change", now i'm confused.

I have another question related with the topic. We know that chain's rule is defines as $'(f(g(x))) = f'(g(x))*g'(x)$. But i can't undestand why is that. I know, there are proofs, but they to hard for me at this moment and i can't read the intuition that stands behind chain's rule.

Thx to every response.

Arlic
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1 Answers1

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The geometric idea is what you stated about the tangent line. The analytic idea is:

$$f(x+\Delta x)=f(x)+f'(x) \Delta x + r(x,\Delta x)$$

where $r(x,\Delta x)$ is small if $\Delta x$ is small. That is, $g(\Delta x) = f(x+\Delta x)$ is nearly a linear function if $\Delta x$ is small enough. "Small" and "small enough" are quantified by the definition. Specifically, if you rearrange the definition of the derivative, you get:

$$\lim_{\Delta x \to 0} r(x,\Delta x)/\Delta x = 0.$$

To think about the chain rule, let's consider a physical situation. Let's say we're moving along a path which is the graph of some function $y(x)$. Then at each time $t$ we are at the point $(x(t),y(x(t))$. The chain rule tells us how to compute the $y$ component of our velocity:

$$\frac{dy}{dt} = \frac{dx}{dt} \frac{dy}{dx}$$

This says that the $y$ velocity is the $x$ velocity times the slope of the curve that we are following at the point where we currently are. Equivalently, we can look at

$$\frac{dy}{dx} = \frac{dy}{dt} \frac{dt}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$

This says that if we follow a curve $(x,y(x))$ with a parametrization $(x(t),y(t))$, the slope of the curve is the ratio of the $y$ velocity to the $x$ velocity.

Ian
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  • For what does $r(x, Δx)$ stands for ? it is just a function that is described as "small enough" if $Δx$ is "small enough" ?

    You said also that "$g(Δx)=f(x+Δx)$ is nearly a linear function if $Δx$ is small enough", do you mean that behavior of $g(Δx)$ is like linear function for infinitely small Δx ? (ie. Value of function changes infinitely small)

     – Arlic Sep 21 '14 at 17:46
  • I'm avoiding infinitesimals here, following the classical treatment; I would suggest you do the same, as infinitesimals usually confuse beginners more than they help. Here $r(x,\Delta x)$ is the error in approximating $f(x+\Delta x)$ by its tangent line at $x$. The "smallness" of this error is made precise by the second equation in my post. – Ian Sep 21 '14 at 17:53
  • @Arlic Maybe an example would be useful. With $f(x)=x^3$ we get $f(x+\Delta x)=x^3+3x^2 \Delta x + 3x \Delta x^2 + \Delta x^3$. So $r(x,\Delta x)=3x \Delta x^2 + \Delta x^3$. Now take a look at this plot. http://www.wolframalpha.com/input/?i=plot+1%2B3%28x-1%29%2Cx^3-1-3%28x-1%29%2Cx%3D0.8..1.2 This graphs the tangent line of $x^3$ at $1$ with the error term (when you add these together you get $x^3$). Note how much smaller the error term is than the tangent line. This is exactly what differential calculus is for. – Ian Sep 24 '14 at 14:44
  • Therefore differential calculus is used for linear approximation only ? If we will continue with the function described with x^3, then the graph http://goo.gl/HvlBGw , gives as a information that between 0.95<x<1.05 the tangent line approximates our function very accurately.

    For chain rule i have considered a intuitive way, could you check it? Consider a function g(t) that describes path of a car and f(x) that describes path of a train that varies on path of car. If we will try to calculate how fast train travel with respect to time, we will conclude intuitively that is .... (i'll continue)

     – Arlic Sep 24 '14 at 19:19
  • that is ... (how many times is train faster from car) times (how fast is car), we know that to describe "how fast travel our car", we will use derivative, and to describe "how many times our train travel faster from car" we will use derivative with respect to car's path. We can even describe how many times our train is faster than car by '(f(g(x)))/g'(x) = f'(g(x))*g(x)/g(x) = f'(g(x)), Therefore it even fits!. – Arlic Sep 24 '14 at 19:34