Give me some examples on differentiation(with answers) so that i understand this concept better?

Question

why do u differentiate at all?

Answer 1

If you just have a constant number (like 1, 2, 3, etc.) its derivative is always 0. If you have an expression in the form x^n you would use the formula n*x^n-1.

So, if you wanted to take the derivative of x^2 you would bring the 2 down in front of the x and then subtract 1 from the exponent. So you'd get 2x^1 which is just equal to 2x.
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Say you wanted to find the derivative of:

this * that

Then you would use the product rule which says to take the (derivative of the first * the second) + (derivative of the second * the first). So in this case the derivative of this * that would be...

(this' * that) + (that' * this)

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Similarly, if you wanted to take the derivative of top/bottom the quotient rule says to do so like this...

[(top' * bottom) - (bottom' * top)] / (bottom)^2

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If you have a composition of functions, like outside(inside) you use the chain rule, which works like this...

1.) Take the derivative of the outside function leaving the inside alone.
2.) Multiply by the derivative of the inside function.

So in the example outside(inside) the derivative would be...

outside'(inside) * inside'

Answer 2

simple examples:
take the derivative of
8 =0
8x =8
8x^2=16x
8x^3=24x^2
8x^4=32x^3
8x^5=40x^4

so basically (8n)x^n-1 gives you the derivative
the problems start when you have more things to differentiate in the problem

why you differentiate is different.
It depends on the application. you might differentiate because you are given a point and need to figure out the velocity or acceleration with respect to time like lets say you're given a position like this
s=8t+5t^2
you would take the derivative of that which would be
s'=10t+8
which would be your velocity and then you would just plug in your time(t)
basically all it is is the rate of change
like change in something with respect to something else.
dv/dt means change in velocity over change in time
like miles per hour which would be the velocity
its the change in miles with respect to the change in hours

Answer 3

Try Google.