hi everyone today we're going to talk about how to find f of x given the functions second
derivative to complete this problem we'll work backwards and take the integral of the
second derivative to find the first derivative and then the integral of the first derivative
to find the original function let's take a look in this particular problem we've been
asked to find f of x if f double prime or the second derivative of x is equal two thirds
x to the two thirds so in order to find f of x the original function from f double prime
of x or the second derivative we need work backwards what we'll do is take the integral
of f double prime of x to find the first derivative f prime of x and then take the integral of
that to find f of x if we have the second derivative f double prime of x equal to two
thirds x to the two thirds then to find the first derivative f prime of x we'll take the
integral of the second derivative two thirds x to the two thirds and when we take the integral
we have to have the corresponding d x notation so to take the integral we'll just use simple
power rule and because we have x to the two thirds here we'll add one to the exponent
so two thirds plus one or we can two thirds plus three thirds will give us five thirds
so we'll get x to the five thirds and then we want to take the coefficient two thirds
and divide it by the new exponent five thirds so two thirds divided by five thirds but instead
of taking a fraction divided by a fraction we can say two thirds and instead of dividing
by five thirds that's the same thing as multiplying by the inverse of five thirds which is three
fifths so that is the integral of the second derivative but of course whenever we take
an integral we have to add c so we'll get plus c here and all we need to do is simplify
notice obviously that the three in the numerator and denominator here will cancel so we'll
get two fifths x to the five thirds plus c and that is f prime of x our first derivative
function now we need to take the integral of that to get back to f of x so f of x will
be equal to the integral of our first derivative function so two fifths x to the five thirds
plus c and of course as always our dx notation now in order to take this integral we'll do
the same thing we did last time we'll add one to the exponent so x to the five thirds
here we add one to five thirds five thirds plus one will just give us eight thirds so
we'll get x to the eight thirds and now we want to take two fifths the coefficient and
divide it by the new exponent so two fifths divided by eight thirds is the same thing
as two fifths times three eighths so we'll multiply them and now c we can treat as a
constant this is basically the same thing as saying c times x to the zero power x to
the zero or anything to the zero power for that matter is equal to one so this is the
same as saying c times one or just c so by multiplying by x to the zero we haven't changed
anything but it now allows us to treat this the same way we did this x to the five thirds
here we'll just one to the exponent so we'll get zero plus one just gives us one and then
we'll divide c by the new exponent so c divided one is what we'll get and because we took
another integral we need to account for the constant of integration and add c again we
want to distinguish it because we already added c so instead of using c we'll use the
next variable and call it d so now we just need to simplify this is much as we can to
get our final answer we'll get f of x is equal to two fifths times three eighths will give
us six over forty which is the same thing as three over twenty x to the eight thirds
plus c x plus d and that's it that's our original function f of x if the second derivative f
double prime of x is equal to two thirds x to the two thirds so I hope you found that
video helpful if you did like this video down below and subscribe to notified of future
videos