Derivative of Polynomial Functions Video Lecture | Crash Course for MAT

in this lesson we're gonna go over
derivatives of polynomial functions so
let's say if we have the function f of X
and it's X cubed minus 5x squared plus
7x minus 4 so what is the derivative of
that function so all you need to do is
find the derivative of each term
separately so using the power rule the
derivative of X cube is going to be 3x
raised to the 3 minus 1 or 3x squared
now using the constant multiple rule the
derivative of 5x squared is going to be
5 times the derivative of x squared
which is 2x to the first power the
derivative of X is simply 1 and the
derivative of a constant is zero now 5
times 2x is 10x so this is the answer
it's 3x squared minus 10x plus 7 now
let's try a similar example so let's say
we have 4x to the fifth power minus six
x cubed plus 8x squared minus 9 so feel
free to take a minute and find the first
derivative of this function so the
derivative of X to the fifth using the
power rule which is n x raised to the N
minus 1 in this case n is 5 so it's
going to be 5x to the 5 minus 1 or 5x to
the fourth the derivative of x cubed is
3x squared and the derivative of x
squared is 2x and the derivative of the
constant is simply zero so this is going
to be 4 times 5 which is 20 6 times 3 is
18 and 8 times 2 is 16 and so it's 20 X
to the fourth power minus 18 x squared
plus it
Phoenix now let's say if we have a
function that looks like this let's say
it's 7x times 2x minus X cube how can we
find the derivative of this function
well would you do if you see an
expression like this one go ahead and
distribute if we distribute 7 X 2 2 X 7
times 2 is 14 X times X is x squared so
we're gonna get 14x squared and then if
we multiply 7 X to negative X cube
that's negative 7x to the fourth power
so now in this form we could find the
first derivative the derivative of x
squared is 2x and the derivative of x to
the fourth is 4x cubed so the final
answer is going to be 28 X and 7 times 4
is also 28 so 28 X minus 28 X to the
cube now let's say if f of X is equal to
3x plus 2 squared what's the first
derivative of this function now if
you're not familiar with the chain rule
it's best to expand this expression so
we can write it twice since its squared
and we need to foil the expression 3x
times 3x is 9x squared and then 3x times
2 is 6x and then we also have 2 times 3x
which is also 6x and then 2 times 2 is 4
and then if we combine like terms we can
rewrite f of X as 9x squared plus 12x
plus 4 so now in this form it's gonna be
very easy to find the first derivative
so if we differentiate x squared it's 2x
the derivative of X will always be 1 and
for the constant it's gonna be 0 so f
prime of X is gonna be 18x plus 12 and
that's the answer
let's work on one more problem so let's
say that f of X is equal to four X to
the fifth power minus five x to the
fourth power plus two x to the third all
divided by x squared what do you think
we need to do for this example the best
thing to do is to simplify before
finding the first derivative and if you
have a polynomial divided by a single
term or a monomial divide each term of
the polynomial by that monomial
so when dividing by two monomials you
need to subtract the exponents 5 minus 2
is 3 so this is going to be 4x to the
third power and then 4 minus 2 is 2 so
that's 5x squared and then 3 minus 2 is
1 so this is going to be 2x so that's f
of X and it's simplified form so now we
can determine f prime of X the
derivative of X cube is 3 x squared the
derivative of x squared is 2x and the
derivative of X is 1 so the final answer
is going to be 12x squared minus 10x
plus 2 so now you know how to find the
derivative of a polynomial function