Whats the difference between the Derivative and Differentiation?

Question

I'm a 15 year old maths student who recently started studying calculus. I can understand some of it but the text books I use really are rather ambiguous and unclear. The studying which I am doing is quite independent meaning i do not have much expert help from anyone. I am figuring it all out my self.
I know that differentiation is the process of finding the gradient of a polynomial at any give x value.
I know that the formula for this is ax^y = yax^y-a and you write dy/dx before it meaning change in y over change in x. Is the derivative what you get from differentiation? Why dont they just write the 'gradient' or the 'dy/dx? Sometimes i also see d/dx. What does this mean? thank you

Answer 1

Matt -



Differentiation is the "process" of finding the derivative of a function. The derivative is simply the "instantaneous" slope at a given point on the curve of a function. For a linear line it is straight forward and always equal to a constant (y = mx + b, so dy/dx = m).



You will find a multitude of symbolism with respect to differentiation ... each author has their own preference.



y' = dy/dx or d(f(x) /dx all mean the same thing ... take the derivative of the function f(x) with respect to x.



Hope that helps and good luck in your studies



P.S. - you might want to try youtube ... it has plenty of good videos on calculus:



http://www.khanacademy.org/search?page_search_query=calculus

Answer 2

You're right that the derivative is what you get after differentiation and some people do say the gradient or the slope or d ...../dx or dy/dx or even f'(x).



The reason for the multiple notation is because there was some controversy over who came up with Calculus both Newton and Leibniz came up with it independently but to recognise both as having a hand in calculus two different methods are used for the notation. The convention is:



If there is an equation y = (some expression involving x)

the derivative is written dy/dx



but if y = x^2, it's just as true to say d x^2/dx since x^2 means the same thing as y in this case.



Now if someone gives the question in the format f(x) = x^2 then it's conventional to express the derivative as f'(x) = 2x



The only thing you got wrong in your question is where you say the derivative of ax^y = yax^(y-a) that's wrong, it's actually yax^(y-1) but otherwise that was very good.



If you're confused about this why don't you watch some of Salman Khan videos about calculus he's really good.



http://www.khanacademy.org/

Answer 3

Yes, differentiation is the process of finding the gradient of a polynomial at any give x value.



Yes, the derivative is what you get from differentiation.



You need not be confused even if different terms are used.



Given y = f(x), there are other notations for the derivative apart from dy/dx, namely:



y'

f '

f ' (x)

d/dx

d(f)/dx

d[f(x)]/dx



All of these notations mean the same thing: the derivative of the function y=f(x).



Also, you need to learn that there's not just one, but quite a few rules of differentiation:



(1) The Derivative of a Constant Function:

Given; y = k, where k is a constant

dy/dx = 0



(2) The Derivative of a Constant Times a Function:

Given: y = k*f(x)

dy/dx = k*f ' (x)



(3) The Power Function Rule:

Given: y = x^n

dy/dx = n * x^(n - 1)



(4) The Product Rule:

Given: y = u(x) * v(x)

dy/dx = u*v' + v*u'



(5) The Quotient Rule:

Given: y = u/v

dy/dx = (v*u' - u*v') / v^2



(6) The Generalized Power Function Rule:

Given: y = [g(x)]^n

dy/dx = n * g^(n - 1) * g'

Answer 4

Derivative is the value of the coefficient of differentiation. It indicates the rate of change in the dependent variable (y) for a unit change in independent variable (x). It can be denoted with f' or dy/dx.

dy/dx means the derivative of the function y with respect to the variable x.

d/dx is simply a sign or notation like = or + or - or sigma. It implies that the derivative is with respect to x.

Depending on the function the letters y and x can be changed.



for ex. If Profit function is given as C(P) = 15x^2 + 3x - 100

then the derivative can be dP/dx which means rate of change in profit for a unit change is x

In the same manner dC/dx is used for for the derivative of cost function, dR/dx is used for the derivative of revenue function so on...



Differentiation is the process of finding the rate of change in dependent variable or the gradient of a polynomial at any given x value.



YES, the quantity what we get from differentiation is the derivative.



Everything depends on the usage in vogue.