Don't listen to the guy above talking about integration. Integration is NOT the reversal of differentiation. That would be the anti-derivative. Integrals and anti-derivatives are NOT the same thing. But they are connected by the Fundamental Theorem of Calculus.

If a function f(x) has an anti-derivative F(x), the area under the curve from a to b is equal to F(b)-F(a).

This is integration defined.

This is integration defined.

by MIT 2010 January 14, 2007

the reverse process of differentiaton

we know that, for example if f(x) = 2x^3 - 5x^2 + 3x -7

then f'(x) = 6x^2 - 10x + 3

This process can be reversed.

In general, y = x^n -> dy/dx = nx^(n-1)

So, reversing this process, it would seem that dy/dx = x^m -> y = (1/(m+1))x^(m+1)

The general process of finding a function from its derivative is known as interation.

we know that, for example if f(x) = 2x^3 - 5x^2 + 3x -7

then f'(x) = 6x^2 - 10x + 3

This process can be reversed.

In general, y = x^n -> dy/dx = nx^(n-1)

So, reversing this process, it would seem that dy/dx = x^m -> y = (1/(m+1))x^(m+1)

The general process of finding a function from its derivative is known as interation.

Given that dy/dx = 12x^2 + 4x - 5, find an expression for y.

y = 12((x^3)/3) + 4((x^2)/2) - 5((x^1)/1)

It would seem that

y=4x^3 + 2x^2 - 5x

but that is not quite the complete answer

Whenever you differentiate a constant you get zero,

e.g. y = 7 dy/dx = 0

and so the expression for y above could have any constant on the end and still satisfy dy/dx = 12x^2 + 4x - 5

The answer to this example is therefore

y= 4x^3 + 2x^2 - 5x + c, where c is a constant.

y = 12((x^3)/3) + 4((x^2)/2) - 5((x^1)/1)

It would seem that

y=4x^3 + 2x^2 - 5x

but that is not quite the complete answer

Whenever you differentiate a constant you get zero,

e.g. y = 7 dy/dx = 0

and so the expression for y above could have any constant on the end and still satisfy dy/dx = 12x^2 + 4x - 5

The answer to this example is therefore

y= 4x^3 + 2x^2 - 5x + c, where c is a constant.

by hotgirl69xxx December 22, 2004

by Atticus Coon March 07, 2005