| title | What is Derivatives ? |
|---|---|
| tags | Calculus |
In algebra you learned to calculate the slope of a straight line. y = mx + b, right? m is the slope.
But what about a curved line? Then the slope isn't constant. It changes: it's higher when the line is steeper, and lower when the line is flatter. So instead of a single number, the slope is a function of x. At any given x, your line has a certain y value, and a certain slope.
Calculus is the set of math tools that allow you to find the slope function from the original function, or the original function from the slope function.
The derivative is finding the slope function. For example the derivative of y = x^2 is y'=2x. That means a basic parabola has a slope of 2x.
Later you will learn about integrals, which are the opposite: they allow you to reverse the derivative, finding the original function from the slope function.
Derivatives allow you to calculate the slope for equations that aren't simplistic and linear... like actual real-world situations.
A more real example where this is useful: say you've got a function that describes the height of a ball thrown in the air. It goes up and down in a parabola. What is the maximum height the ball reaches? Well, calculate the derivative, and solve that for zero, because at it's peak, the "slope" of the parabola will be zero. Take that time you just solved for, plug it back into the original function, and you've got your maximum height.
When you start learning functions, you focus on what the value of the function is at some point. "At x=10 what is y=f(x)?" It's also useful to know things like "at x=10, how is the function changing?". That's where derivatives come in. If it's a 2D plot, the derivative at that point is its exact slope. So we know how steep it is and if it's increasing, decreasing, or flat there
Since you're in a calc class, let me give an example of what you'll encounter. If you have some function, like f(x)=x2, you'll take the derivative and get f'(x)=2x, where f'(x) is the derivative of f(x). I can evaluate f(x) to get the value of the function at some x value. If I do the same thing with f'(x), instead of the value of the function, I get the slope of a tangent line touching the function at that point. So for x=3, f'(3) = 2x3 = 6. So the slope of the parabola at that point is 6. If I do some quick substitution (don't worry, you'll get to this later in the class, I'm just demonstrating), I can get that tangent line as y=6x-9 (notice the slope of the line is 6), which I have plotted here for your pleasure.
A derivative tells you how fast something changes with respect to something else. Example: say you turn on a heater in your room and the air temperature starts to rise. The 'derivative of temperature with respect to time' (dT/dt) is how quickly the temperature is rising/falling at each moment. At first, dT/dt will be positive, which means the temperature is increasing with time. After some time, the temperature will stabilizie, so dT/dt = 0 (there is no change in temperature as time goes on). Finally, if you turn the heater off, the temperature will begin to drop, so dT/dt is negative (temperature is decreasing with time).
But you can take derivatives with respect to anything. You could take the derivative of the temperature with respect to the distance to the heater, dT/dx (meaning, how does the temperature vary as I move further away from the heater?) which will be negative, since the temperature decreases as you increase the distance to the heater.
The derivative of a function is the instantaneous rate of change of that function.
Let's say you are looking at position S of an object as a function of time T. So S(t) = 3t (for example.) You know that T = 0, the object is at S = 0. At T = 1, the object is at S = 3. At T = 2, the object is at S = 6. The rate at which the object's position is moving, its velocity is constantly 3 units per second.
But what if the object wasn't at a constant velocity? What if S(t) = 3t² + 2t?
The derivative lets us find the velocity of the object at any time T, because if S(t) = 3t² + 2t, and velocity is the derivative of position, we know V(t) = 6t + 2. So at T = 0, the object's velocity is 2. At T = 1, its velocity is 8.