Question

I am a bit confused on a derivative is in Calculus. From what I understand, I derivative is the limit of a tangent line on a curve. I was wondering if someone can explain in simple terms what a derivative is and what is the point of calculating it. Thanks!

Answer 1

Answer:

Step-by-step explanation:

To answer your question fully can literally take up an entire  book.

The importance becomes obvious if you consider the derivative (and the inverse  of it the integral) as the foundation of calculus. I guess that askes the question what is calculus? For this answer a very very simplistic answer would be that it is the branch of mathematics that studies "non-constant" change (things that curve)

...... Let's see if  this makes sense? You are driving your car at a constant  50 mph . after one hour you are 50 miles from home and after two you are 100.

y=50x. you can calculate this value in a several of ways just take two time stamps and two distance stamps ... subtract the two times and two

ok now here is the REAL PROBLEM issue we want to know our CHANGE IN

VELOCITY (that is  ACCELERATION) ... if you take ANY two points you will

see that the SPEED is ALWAYS the same so the numerator is always 0 (zero) and thus the denominator (the difference in the time stamps) does not matter

now lets say that we are accelerating ..we are going y = 10 $x^{2}$ mph

at 3 seconds we are going 90 mph (10* 3^2) .. but we really don't care about that   we want the "acceleration" at EXACTLY that point...

to do that we "have to estimate" delta Y over delta X

lets take 3 and 0  .... (9-0)/3 estimate of acceleration is 3...

but WAIT we can do better (get closer) take 3 and 1 ... (9 - 1)/2

acceleration is 4... once again lets get closer to the exact 3 ...

3 and 2  ... (9 -4)/2 the acceleration is 5...

hold on !  lets get even closer use 3 and 3 and we get it exactly !

(9-9)/0 that is the answer hurray!  HOLD ON !  you CAN NOT DIVIDE BY ZERO ... no can do... we have to wait for newton and Leibniz to  

figure out lim as x-> 0 f(x)-f(x+ delta x) / x to get around the DIVIDE BY ZERO problem...

that is the derivative