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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

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marked as duplicate by Don Thousand, Lord Shark the Unknown, Song, mrtaurho, uniquesolution Mar 15 at 9:32

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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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.

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    $\begingroup$ In general, the instantaneous rate of change in any quantity that varies continuously (or close enough to continuously) over time $\endgroup$ – Zubin Mukerjee Mar 15 at 3:55
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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.

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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, $B(x+\triangle x,y + \triangle y)$. ![enter image description here 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 $x$. $$y=mx+c$$ $$\triangle y=\triangle {mx +c}=\triangle {(mx)} + \triangle c= m\triangle x$$ 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 : $$\frac{\triangle y}{\triangle x}=\frac{m\triangle x}{\triangle x}=m$$ 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 : $$\lim_{\triangle x\rightarrow 0}\frac{\triangle y}{\triangle x}=\frac{dy}{dx}=m$$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 $f(x)$ enter image description here 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.....

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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 $\theta’’+\frac g l \sin\theta=0$ 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.

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Differentiation is so widely used that it's almost difficult to know where to start. I'd go as far as to say that it's difficult (but perhaps possible) to find any broad scientific endeavour that doesn't use derivatives at all.

Standard techniques such as least squares regression uses derivatives to achieve its solution. So any time you fit a regression line to some (noisy) data, you're most likely using derivatives (perhaps unknowingly). Least squares regression is used in the "softer" sciences such as sociology and psychology as well as the "harder" sciences such as physics.

Derivatives are widely used in optimization (In fact, the least squares problem is an optimization problem). See this link for some real world applications of optimization. A very basic example of an optimization problem is the following. How do we find the maximal volume a box can contain if it its exterior is made of $1\,\text{m}^2$ cardboard and it must have at least a height of $x$? The process to answer such a question will often use derivatives.

Derivatives are used in differential equations. Differential equations are used to model extremely many physical phenomena and widely applied to solve physical problems, but also non-physical problems. See this link for some real world applications of differential equations. It includes economic models, fluid dynamics models, electrodynamics, and much, much more.

Derivatives are widely used in machine learning. See this link for some applications of machine learning. The list is vast. Applications of machine learning are often related to optimization.

I would also recommend reading a bit about the history of calculus, see specifically applications, because that will give you some insight into why differentiation (and related notions) were invented. For example, Newton's (second) law states that $$F = ma $$ but the acceleration, $a$, uses derivatives because $$a = \frac{dv}{dt} = \frac{d^2x}{dt^2 }$$ Acceleration is the instantaneous rate of change of velocity ($v$), or the second derivative of position ($x$) with respect to time. So we use differentiation to understand motion of objects.

Derivatives were (and are) used in investigation of vibrating chords, on analysis of elasticity of bodies, heat diffusion, electricity, etc. These are generally applications of differential equations as discussed above.

Happy reading!

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Derivatives help us answer the question "How does something change with time?" $\frac{d}{dt}$(something)

Once you realize that something can be replaced with most every phenomena from celestial motion to stock prices, you start recognizing that calculus forms the backbone of all modern science and technology.

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