in this lesson we're going to go over derivatives of polynomial functions so let's say if we have the function f of x and it's x cubed minus 5 x squared plus seven x minus four 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 cubed 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 0. 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 four x to the fifth power minus six x cubed plus eight x squared minus nine 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 one in this case n is 5 so it's going to be 5 x to the 5 minus 1 or 5 x 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 four times five which is twenty six times three is eighteen and eight times two is sixteen and so it's twenty x to the fourth power minus 18x squared plus 16x now let's say if we have a function that looks like this let's say it's 7x times 2x minus x cubed how can we find the derivative of this function what would you do if you see an expression like this one go ahead and distribute if we distribute 7x to 2x 7 times 2 is 14 x times x is x squared so we're going to get 14x squared and then if we multiply 7x to negative x cubed that's negative 7 x 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 28x and 7 times 4 is also 28. so 28x minus 28 x to the cube now let's say if f of x is equal to three x plus two 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 it's 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 going to be very easy to find the first derivative so if we differentiate x squared it's 2x the derivative of x will always be one and for the constant it's going to be zero so f prime of x is going to 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 4x to the fifth power minus 5x to the fourth power plus 2x 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 monomial divide each term of the polynomial by that monomial so when dividing by two monomials you need to subtract the exponents five minus two is three so this is going to be four x to the third power and then four minus two is two so that's five x squared and then 3 minus 2 is 1. so this is going to be 2x so that's f of x in its simplified form so now we can determine f prime of x the derivative of x cubed is 3x squared the derivative of x squared is two x and the derivative of x is one so the final answer is going to be 12 x squared minus 10x plus 2. so now you know how to find the derivative of a polynomial function you