What is the significance of the slope of the tangent line of a function? Why is the derivative so important?

As I finished calc 1. I can use the product rule and chain rule and resolve integrals. But I feel like its too mechanical for my taste. I know the procedure and I execute on paper without really understanding or experiencing the "ahaa moment".

For example, when I was learning geometry in elementary school, the "ahaa moment" for me was when I had to move furniture in my room and needed to find areas of stuff. I'm trying to find the equivalent application of the derivative and integral as I learn calculus. Could someone demystify this?

Thank you in advance.

A slope is a rate of change. If a car moves at 30 miles per hour, and you graph its distance traveled as a function of time, the slope of that graph is 30 miles per hour.

Isaac Newton was the first person ever to show that the laws of nature that govern objects we see every day here on Earth are the same as the laws of nature that govern things in the heavens: planets, moons, comets (including the planet Earth as a whole).

In order to do that he had to think about the rates at which they move as a function of their locations.

Rates of change also occur in the following way: Newton asked himself whether the gravitational pull of the earth on the moon or other orbiting object is the same as if the whole mass of the earth were concentrated at the center. (The bottom line, after much effort, turns out to be "yes".) So how do you find the sum of the infinitely many infinitely small quantities involved? The answer is that you ask how fast that sum changes as progressively more of the earth is taken into account. This is not a rate of physical motion as time passes, but is something else.

Unfortunately far too much of the conventional mathematics curriculum is designed to prepare students for later courses, in such a way that you find out the motivations only if you take certain later courses that most of the students will never take. It leaves students not understanding the whole thing. Honest education does not do that. A few professors here and there are working on calculus courses that are in that respect honest. At least two professors at Macalester College have been working on that, and at various other places that I can't name right now.

The derivative might just be the most important thing in mathematics. In physics, if the function that gives the motion of a particle with respect to time is defined as $x(t)$, then the derivative of this function $$x'(t)=v(t)$$ gives the velocity of the particle at time $t$. Also, the derivative of the function $v(t)$, or $$v'(t)=a(t)$$ gives the acceleration of the particle. Even Newton's famous law $F=ma$ can be rewritten as the derivative of the momentum with respect to time, or $$\Sigma{F}=\frac{dp}{dt}.$$ Also, the power exerted by a force is equal to the derivative of the work with respect to time $$P=\frac{dW}{dt}.$$ In fluid mechanics, the definition of pressure is given as the derivative of the force with respect to the area that the force is acting upon, or $$p=\frac{dF}{dA}.$$ In physics the list goes on and on, there would be no physics with calculus. The derivative can also be applied inside of mathematics, given the gradient of any curve, which is useful, and the second derivative giving information about concavity and points of inflection.

Basically, it is really important, and if you go into the sciences or engineering as well as studying mathematics, you will notice the derivative, and all other aspects of calculus.

There are two other ways to think of the derivative:

1. as the "instantaneous rate of change" of the function -- that is, how much it is changing at one particular moment were it viewed as a function of time. Namely, the rate of change of a quantity over an interval is $$\mathrm{rate\ of\ change} = \frac{\mathrm{change\ in\ quantity}}{\mathrm{interval\ over\ which\ change\ occurs}}$$ or, in symbols, $$R = \frac{\Delta f}{\Delta x}$$. Then the derivative is the limiting value of this rate when $\Delta x \rightarrow 0$, that is, in the "ideal" case where the interval is a single instant of time. We denote this by $f'(x)$ or $\frac{df}{dx}$, the latter calling to mind the ratio above.

2. as a "best linear approximation" of the function. That is, the derivative tells you what the linear function is which is "closest" to the function at a given point in that every other linear function will have a worse error for points near the point of differentiation. This is what the tangent line represents -- note that it too is really a function. If you've done a little linear algebra, you might have heard of a "linear transformation", which on the real numbers is a function of the form $g(x) = ax$. You can think of this as finding the linear transformation (the $a$-value, more precisely), translated to be relative to the point at which you differentiate, which best approximates the deforming action of the function on a small piece of the real number line about the point at which you take the derivative. Since the linear transformation shown can be thought of geometrically as a "stretch" or "scaling" (that's what it means when you multiply a real number by another real number -- you scale one number by another), you can also think of the derivative as telling you how much the function "stretches" or "scales" a small piece of the real number line about the point of differentiation. This is actually a good concept to keep in mind for when you get to more advanced forms of math, where generalizations of the derivative to more exotic kinds of spaces than the Real Numbers will appear, and it is this sense in which it is used.

Usually in physics, one is most often interested in interpretation 1), whereas in more advanced forms of math, one may be more interested in interpretation 2).

Taking your example of the furniture in the room we can think of the integral as the area under the curve. We divide the area up into small blocks and fit them in there like the furniture. Consider the integral $$ \int_0^1 x^2 dx = {1 \over 3} $$ we can think of this as the area beneath this graph. If we were to grind up the unit square so we could fit small pieces of it into this area we could fit $1\over3$ of a piece.