Tjyylli

I have just started learning about differentiation. I know that differentiation is about finding the slopes of curves of functions and etc.

I have many saying that differential and integral calculus are important tools of math and have many real-life applications.

So I want to how differentiation, which about finding slopes of functions, is important in real life. Explain it with some applications.

I am sorry if this is duplicate of other question. But as far as I have searched, I have not found an answer to this question

The first real world application that comes into mind to me is calculating your "instantaneous speed", or perhaps more accurately, calculating your average speed over an infinitesimally small amount of time.

I'm sure you know that your average speed is given by $frac{text{distance traveled}}{text{time traveled}}$. So, what differentiation gives you is the ability to make the "time traveled" quantity very very small (tending to zero), giving you your speed at a given instant.

If you think about it, this is quite useful, especially if your'e driving a car and would like to know your current speed.

An important application
is finding the
extreme values
(min or max) or a function.

An example:
To find the angle
at which a thrown ball
goes the farthest,
derive the formula
for the distance traveled
as a function of the angle
and then differentiate it
and find the angle
which makes the derivative zero.
That is the angle with
maximum distance.

Differentiation of any function gives you the slope of the tangent at that point. Why? Consider a line with the equation:
$$y=mx+c$$
Consider two points $A(x,y)$ and another point that is slightly far from A, $B(x+triangle x,y + triangle y)$.

where $triangle y$ is the change in the function value for a change corresponding to $triangle x$. In other words, the change in $y$ occurs due to a change in $x$.
$$y=mx+c$$
$$triangle y=triangle {mx +c}=triangle {(mx)} + triangle c= mtriangle x$$
So, now we calculate the slope of the line $AB$, the coordinates are $A(x,y)$ and $B(x+triangle x,y+triangle y)$. The slope is :
$$frac{triangle y}{triangle x}=frac{mtriangle x}{triangle x}=m$$
If you make the changes $triangle x$ and $triangle y$, very small tending to $0$ then the values change to $dy$ and $dx$. The slope is :
$$lim_{triangle xrightarrow 0}frac{triangle y}{triangle x}=frac{dy}{dx}=m$$The visualization is like you bring the coordinate $B$ closer and closer to $A$ and keep on calculating slope until the point $B$ is so close that it coincides with the point $A$. Now you get the slope of the line joining $A$ and $A$ or in other words slope of the line at point $A$.

Same things can be generalized for a function $f(x)$

Consider a function $f(x)$ and a point $A(x,y)$ on it. We take another point $B(x+triangle x,y+triangle y)$. Now we calculate the slope again of the line $AB$ as:
$$frac{f(x+triangle x)-f(x)}{x+triangle x -x}=frac{f(x+triangle x)-f(x)}{triangle x}$$

Now, the line $AB$ is actually a secant to the curve $f(x)$. Just bring the point B closer and closer to A that is keep decreasing $triangle x$ so that it becomes very small. Finally as $triangle x rightarrow 0$, the slope changes to $frac{dy}{dx}$ and the line $AB$ which was a secant back then has turned into a line $AA$ which means that it is a tangent at point $A$. That is why the slope $frac{dy}{dx}$ gives the slope of the tangent at any point.
Finally,
$$lim_{triangle x rightarrow 0 }frac{f(x+triangle)-f(x)}{triangle x}=frac{dy}{dx}$$

This is also known as the first principle or Ab Initio method to compute the derivative.

What else can you do once you know the slope of the line using differentiation. Anything and literally everything. Knowing the slopes of line will tell you the change in the slope is positive or negative giving the idea whether the function itself is increasing or decreasing function. Also if the slope of a tangent becomes $0$ at a point then there is a possibility of the maxima or minima at the same point. You can find the rate of change of quantity with respect to another like velocity or acceleration. The underlying concept is only one, the slope of the tangent.
Hope this helps.....

Derivatives are important in modeling physical phenomena via differential equations. These equations can model real-world processes much more accurately than a purely algebraic one. For example, the equation $theta’’+frac g l sintheta=0$ models the motion of a simple pendulum, where $theta$ is the angular displacement, $l$ is the length of the pendulum, and $g$ is the acceleration due to gravity.