Question regarding Higher Derivatives

[ggcheck]

when working with higher derivatives each time we differentiate a polynomial the degree decreases by 1.

I am asked to find the second and third derivatives of this polynomial: y=(x^4)(e^x)

I get the following (pretty sure it's right):

y`= (x^4)(e^x) + 4(x^3)(e^x)

y`` = (x^4)(e^x) + 8(x^3)(e^x) + 12(x^2)(e^x)

y``` = (x^4)(e^x) + 12(x^3)(e^x) + 36(x^2)(e^x) + 24(x)(e^x)

I thought the degree's of the polynomials were supposed to decrease with each differentiation? Isn't each polynomial still degree 4?

Where is my error

 

[G01]

Your derivatives are right. The problem is that you are not dealing with polynomials. The e^x term is exponential, not polynomial, in nature. So, the properties of polynomials do not necessarily apply here, since your polynomial term in the function is multiplied by and exponential term.

 

[ggcheck]

Your derivatives are right. The problem is that you are not dealing with polynomials. The e^x term is exponential, not polynomial, in nature. So, the properties of polynomials do not necessarily apply here.

arg, I guess I was thrown off by the x^4... that made me think I was dealing with a polynomial. I guess I need to review basic algebra to remind myself what a polynomial is