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

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 mixing of blacks and whites during the 30's and 40's

Integration sucked when Elvis was around

by Atticus Coon
March 07, 2005