The derivative of a polynomial function involving multiple linear and/or power functions can be found using the formulas for finding linear and power functions, along with the constant multiple and sum rules. The constant multiple rule lets the polynomial derivatives of multiples of power functions simply be multiples of their derivatives, while the sum rule allows monomial parts of the polynomial to be calculated one by one.
Lets talk about the derivatives of polynomial functions, now we're going to need 2 properties of derivatives before we can differentiate any polynomial function. And these are the properties. First the constant multiple rule here is how it works, now let's say that you have a function of f of x and you know how to differentiate it. If you have a constant in front of it like say it's 5, 5 times f of x the derivative of that is going to be 5 times the derivative of f of x. Effectively you can move that constant outside of the derivative and just take the derivatives of the inside function f of x.
The sum rule, the derivative of sum of 2 functions f and g is equal to the sum of the two derivatives so you can separate the sum, right? This is almost like distributing the derivative over a sum but dont think of it that way it's just the derivative of a sum is the sum of derivatives. Now those 2 rules we can differentiate any polynomial function and here are some examples. Starting with a really simple one this is actually the power function, the derivative with respect to x of 2 x of the fifth. So first we can use the constant multiple rule to pull this constant 2 outside. So it's 2 times the derivative of x of the fifth. And we dont know how to differentiate power functions the derivative with respect to x of x to the fifth is 5x to the fourth. So this is 2 times 5x to the fourth and our answer is 10x to the fourth.
Another example the derivative with respect to x of x cubed plus ax, well here I use the second property the sum rule that this is going to equal the derivative of x cubed plus the derivative of 8x. Now the x cubed that's just the power function and this derivative is going to be 3x squared. 8x is a linear function, we differentiated these earlier when we know that the derivative of a linear function is just its slope in this case 8. And that's our answer 3x squared plus 8.
Finally let's take a look at this guy, the derivative with respect to x of x to fourth minus 3x squared so again we're going to use the sum rule and the constant multiple rule. This is the same as x to the fourth plus negative 3 times x squared. So this is the derivative with respect to x of x to fourth plus the derivative with respect of x of negative 3 times x squared. Often when I'm first teaching these rules I like to use one at a time so what I've used here is the sum rule and now I'll use the constant multiple rule to pull this negative 3 outside. So just copy that down and I'm going to have plus negative 3 times the derivative of x squared and now I'm ready to finish it up. So this is just the derivative of the power function x to the fourth 4x cubed and add negative 3 times the derivative of x squared which 2x. So thats negative 3 times 2x, and it ends up being 4x cubed minus 6x thats it you just need 2 properties, the constant multiple rule and the sum rule to differentiate any polynomial.