In this problem, we're asked to find f double prime of x or the second derivative. Now our function that we're given here is f(x) = 4x^{-3} + 3x^7 . Now remember that in order to find the second derivative, that just means that we need to take the derivative twice. So let's start by just taking the derivative once, and then we can find it that second time to get our final answer. Now to find our first derivative here, f'(x), I am, of course, going to use the power rule since I have these powers happening here.
So for that first term, 4x^{-3}, I'm gonna pull that negative 3 out to the front, multiplying by it and then decrease that power by 1. So that negative 3 times 4 will give me a negative 12, and then I have x^{-3 - 1} = x^{-4}. Then for that second term, I have 3x^7. Again, using the power rule here, I'm gonna pull that 7 out to the front, multiplying by it, and then decrease that exponent by 1. So that 3 times 7 gives me a positive 21, and then I have x^{7 - 1} = x^6.
Now we're not done yet because this is just our first derivative, and we want to find our second derivative f''(x) by taking the derivative one more time. Now taking the derivative of this function here again using the power rule again just for the second time. I'm gonna pull that negative 4 out to the front, multiplying by it and then decrease that power by 1. Now here, since I have negative 12 times negative 4, that's going to end up giving me a positive 48. So I have 48 x^{-4 - 1} = 48x^{-5}.
Then for that second term, using the power rule again, pulling that 6 out to the front, multiplying it by it, that will give me 126. And then decreasing the power of that x by 1, that will give me x^{5}. And this is my final answer here. The second derivative, f''(x), is 48x^{-5} + 126x^5. Let me know if you have any questions here.
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