Define a Derivative:
We know that the derivative tells us about the slope of any graph at a given point, which we have learned in our previous concepts of derivatives.
Consider some function f(x)
The derivative can be interpreted as the slope of this graph above some certain point right ?
In terms of slope we can define a derivative in two ways. A steep slope means a high value for the derivative, a downward slope means a negative derivative.
So, what's the second derivative ?
Second Derivative:
We can say that the second derivative is the derivative of a derivative. Means it tells us how the slope is changing.
The way to see that at a glance is to think about how the graph of f(x) curves at points where it curves upwards the slope is increasing and that means the second derivative is positive where it's curving downwards the slope is decreasing, so the second derivative is negative.
Second Derivative Notation:
The notation of first derivative is the main idea behind the second derivative notation, which is
$$ \frac{dy}{dx} \;\;\;or\;\;\; f'(x) $$
Similarly , it should be written as:
$$ \frac{d(\frac{dy}{dx})}{dx} $$
You could try writing it like this, which indicating some small change to the derivative function divided by some small change to x, where as always the use of this letter “x” suggests that what you really want to consider is what this ratio approaches as dx both the “x's” in this case approach zero.
That's pretty awkward and clunky so the standard second derivative notation to abbreviate is:
$$ \frac{d^2y}{dx^2} or f''(x) $$
It's not terribly important for getting an intuition for the second derivative but it might be worth showing you how you can read this notation.
Representing as an Acceleration Derivative of Velocity:
The most visceral understanding of the second derivative is that it represents acceleration.
Given some movement along a line,
Suppose you have some function that records the distance traveled versus time. Maybe its graph looks something like this is steadily increasing over time
Which is known as displacement (s), means change in position by some object.
Then the first derivative tells us the change in displacement with respect to time which is called as velocity at each point in time,
$$ \frac{ds}{dt} $$
For example the graph might look like this bump increasing up to some maximum and then decreasing back to zero.
So the second derivative tells you the rate of change for the velocity which is represents an acceleration derivative of velocity at each point in time.
In this example the second derivative is positive for the first half of the journey which indicates speeding up.
That's the sensation of being pushed back into your car seat or rather, having the car seat push you forward a negative second derivative indicates slowing down, Negative acceleration.
Second Derivative Rules:
The second derivative is the derivative of first derivative or we can cal the "derivative of a derivative". There are not much complex rules for second derivative, only we have to find the derivative of first derivatove function. To learn how to find it, these are two basic second derivative rules which are given below:
- Take the derivative of function.
- Then take the derivative of that derivative.
Let’s elaborate this concept by solving dimple examples.
Example:
Now the time to test what we learn in second derivative. So let's solve an example using second derivative rules.
Consider we have a function:
$$ y = 3x^4 $$
Solution:
The first derivative of function will:
$$ \frac{dy}{dx}= 12x^3 $$
Now, again take derivative for second derivative of that function:
$$ \frac{dy}{dx}=36x^2 $$