Introduction to Derivatives
We know that the derivative is the instantaneous rate of change. Most of us think about where we can use derivatives in our daily life.
But if you think geometrically then there are a lot of cases where you find derivative examples in day-to-day activities. The biggest use of derivatives calculus is Optimization as all of us need the best for ourselves.
Like someone wants the highest grade without study or someone wants to earn more while sitting at home. Means everyone needs the best and cheapest way of doing things and here our derivatives in the mean of geometry take place.
For example, if someone wants to launch his new product then he should need to know how to price for getting more audience.
Similarly, if someone has a business of producing cylindrical cans(tins) then the question is he needs the maximum cylindrical volume by using the same tin then how much height and radius will be required?
So the geometric meaning of derivatives can help in such types of problems.
Derivative as a Slope:
We will try to elaborate the geometry meaning of derivatives by a simple example.
Let’s say we have a ladder which we place in two different ways.
Probably people find differences in their slope or maybe in angles as to θ1 and θ2. If we have to calculate their angles then it quite easy using
$$ tanθ = \frac{Rise}{Run} $$
If you imagine a ladder on the y-axis and x-axis then,
$$ tanθ= \frac{Δy}{Δx} $$
This means a change in height with respect to a change in distance. If you notice this is the same as the definition of the average rate of change. In general
$$ Slope = \frac{Δy}{Δx} = Avg\;rate\;of\;change $$
So, we can extend that idea if we have any graph given for calculating the average rate of change between any two points of a graph then we can use the same concept.
We have to simply find the slope between these two points in any type of graph either straight or curved, and that slope will be the average rate of change.
Similarly, finding the slope between any two points is called the slope of secant. You can simply write as
$$ Slope\;of\;Secant= Avg\;rate\;of\;change $$
In the same situation, the slope between any two points in a straight line will always be the same because the average rate of change is the same. It means for any instance between the two-point rate of change will be the same.
In general, if we wish to find the derivative of the instantaneous rate of change of any straight points we simply calculate the slope of any two points of that straight line.
So straight line has two special cases as shown in the figure:
- Along the x-axis, in which the straight line is parallel to the x-axis, in this case, the slope between any two points will always be zero. So if the slope is zero it means the derivative of that line is zero.
- Along the x-axis, in which the straight line is parallel to the y-axis, in this case, the slope between any two points will always be undefined. So if the slope is undefined it means the derivative of that line is undefined.
Now let’s talk about slope for the curve graph, in which the slope between any two points is different. So if the slope is different it means the instantaneous rate of change or derivative is different.
Now for calculating slope for any single point, can we use the same derivative formula? Definitely not.
Now we see how to find the slope at any single point in a curve graph.
Derivative at a point in a curve:
First of all, we have to keep in mind that derivative as a slope or for straight or constant things is not enough. The real meaning of calculus is changing which always occurs in our daily life.
So we have to find the derivative at a point. Let’s take an example, we have a graph as shown below
Here you can see, for the slope of points A to B, you can draw a straight line and then calculate the rate of change between them. But how to find the slope at a single point A is the main question.
For such conditions, we use the same method as we use in the concept of derivatives calculus, we try to take B near to A and take them closer to each other so that their distance is approximately near to zero. In that situation, that line touches the curve at one point which is point A.
In mathematics, when any line touches another line at a single point it is called a tangent. So, the slope of tangent is the slope of that single point
I.e.
$$ Slope \;at\; any\; point = Slope \;of \;tangent $$
This point gives you an instantaneous rate of change at that point means the derivative at a point.
Derivative at a point:
We can find the derivative of a function y=f(x) by using a slope of tangent formula:
$$ Slope= \frac{Δy}{Δx} $$
Here we know delta indicates a change in y and x as we can see in a graph given below:
From a graph, we can observe that
We have to solve
$$ \frac{Δy}{Δx} = \frac {f(x+Δx) − f(x)}{Δx} $$
And also approaches Δx towards 0, so
Let’s see derivative examples for proper understanding
Example:
Consider a function:
$$ y=x^2 $$
Solution:
$$ f(x)= x^2$$
So, we can find x+Δx
$$ f(x+Δx)= (x+Δx)^2 $$ $$ f(x+Δx) = x^2 + 2xΔx + (Δx)^2 $$
Now, put it in the slope formula
$$ \frac{Δy}{Δx} = \frac{f(x+Δx) − f(x)}{Δx} $$ $$ \frac{Δy}{Δx} = \frac{x^2 + 2xΔx + (Δx)^2-x^2}{Δx} $$ $$ \frac{Δy}{Δx} = \frac{2x Δx + (Δx)^2}{Δx} $$ $$ \frac{Δy}{Δx} = \frac{Δx(2x + Δx)}{Δx} $$ $$ \implies \frac{Δy}{Δx} = 2x +Δx $$
As, Δx approaches towards 0.
$$ \implies \frac{Δy}{Δx} = 2x $$
So, the slope of x2 at point x is 2x.
Or in general, we can say that the derivative of a function f(x)= x2 is 2x.
Derivative Formula:
For finding derivative we know we have two things required:
- The slope of the function as Δy/Δx.
- Approaching Δx towards 0.
We have a concept of limits approaching 0. So, we can convert Δx towards 0 in terms of limit. In this way,
$$ \frac{Δy}{Δx} = \lim\limits_{ Δx \to 0} \frac{f(x+Δx) − f(x)}{Δx} = f'(x) $$
And the standard notation for differentiation formula will:
$$ \frac{dy}{dx} = \lim\limits_{ Δx \to 0} \frac{f(x+Δx) − f(x)}{Δx} $$
This is called a derivative of a function or differentiation.