calculus. need help:)?

Question

find the second order derivative and evaluate at x=3

y = 10x^3 + 8x^2 + 19

f(x) = (4x - 1)(3x^2 + 2)

find the successive derivatives and evaluate at x=2

f(x) = 2x^3 + 7x^2 + 9x - 2

f(x) = (8 - 2x)^4

please help me with that lol:) ..id really, really appreciate it :)

Answer 1

For the first and third problems, you just need to know how to differentiate a polynomial function. To differentiate a polynomial, notice that it's just a sum of terms, each of which is of the form constant*x^n. Of course, the derivative of the sum is just the sum of the derivatives. So differentiate each c*x^n term, and add up the results. You'll get another polynomial, so the derivative of a polynomial is another polynomial, and so is the second derivative, and so on. (Differentiate often enough -- specifically, one more time than the degree of the polynomial -- and you eventually get to the zero function.)

For the second problem, you have a product of two polynomials. So use the Product Rule to get you started, then differentiate each of the polynomials you're supposed to, and finish by simplifying what results.

For the fourth one, you need to use the Chain Rule. Let g(x) = x^4 and h(x) = 8-2x. You're differentiating g(h(x)).

And if you don't remember the derivative of c*x^n, where c and n are constants -- well, it's c * n * x^(n-1). ;)