Find f"(x) for the function.
f(x)= x^2+√x
f(x) = x^2 + √x
We can rewrite this as:
f(x) = x^2 + x^(1/2)
Therefore, the first derivative is
f'(x) = 2x + (1/2)*(x^(-1/2))
And the second derivative is
f''(x) = 2 + (-1/4)*(x^(-3/2))
Hope this helps :3
f(x) = x^2 + x^(1/2)
f ' (x) = 2x + (1/2)x^(-1/2)
f '' (x) = 2 - (1/4)x^(-3/2)
or 2 - 1/(4x√x)
explanation:
(x^(-3/2)
= 1/x^(3/2)
= 1/(x^(1/2)^3
= 1/(√x√x√x)
= 1/(x√x)
from the ans choices i have, I choose 8x^(3/2)-1/4x^(3/2)
even though both Jai and I had the same answer, which is different from the one you picked ?
Where would the 8 even come from ????
haha is it then 2x^(3/2)-1/x^(3/2)
I was doing final review for my exam and the ans to this question is given 8x^(3/2)-1/4x^(3/2) at the end of the answer key sheet. i might have to check with my prof. But thank you Reiny & Jai.
To find the second derivative, denoted as f"(x), of the function f(x) = x^2 + √x, we need to take the derivative of the first derivative of f(x). Here's how to do it step by step:
Step 1: Find the first derivative, f'(x):
To find the first derivative of f(x), we differentiate each term separately. The derivative of x^2 with respect to x is 2x, and the derivative of √x with respect to x is (1/2) * x^(-1/2). Therefore:
f'(x) = 2x + (1/2) * x^(-1/2)
Step 2: Find the second derivative, f"(x):
To find the second derivative, we differentiate the first derivative function, f'(x), with respect to x. We treat each term as a separate function that needs to be differentiated. Differentiating 2x with respect to x gives us 2, and differentiating (1/2) * x^(-1/2) with respect to x gives us (-1/2) * (1/2) * x^(-3/2), simplified as -x^(-3/2)/4. Therefore:
f"(x) = 2 - x^(-3/2)/4
So, the second derivative of f(x) = x^2 + √x is f"(x) = 2 - x^(-3/2)/4.