if f'(x)=2(3x+5)^4 , then the fifth derivative of f(x) at x=-5/3 is

A. 0
B. 144
C. 1,296
D. 3,888
E. 7,776

I got A

the 5th derivative of a 4th degree polynomial is zero.

But since they gave you f' rather than f, the 5th derivative of f is the 4th derivative of f'.

SO, f^(5) = 2*3^4 * 4! = 3888

To find the fifth derivative of the function f(x), we need to apply the power rule repeatedly.

First, let's find the second derivative of f(x) by applying the power rule to f'(x):

f''(x) = [2(3x + 5)^4]' = 4(2)(3x + 5)^3(3) = 24(3x + 5)^3

Next, let's find the third derivative of f(x) by applying the power rule to f''(x):

f'''(x) = [24(3x + 5)^3]' = 3(24)(3x + 5)^2(3) = 216(3x + 5)^2

Now, let's find the fourth derivative of f(x) by applying the power rule to f'''(x):

f''''(x) = [216(3x + 5)^2]' = 2(216)(3x + 5)(3) = 1,296(3x + 5)

Finally, let's find the fifth derivative of f(x) by applying the power rule to f''''(x):

f'''''(x) = [1,296(3x + 5)]' = 1,296(3) = 3,888

Now, we want to evaluate the fifth derivative at x = -5/3:

f'''''(-5/3) = 3,888

So the fifth derivative of f(x) at x = -5/3 is 3,888.

Therefore, the correct answer is D. 3,888.

To find the fifth derivative of f(x) at x = -5/3, we need to differentiate the given function repeatedly. Here's how you can do it step-by-step:

Step 1: Given f'(x) = 2(3x + 5)^4, we need to differentiate it to find f''(x). To differentiate a function of the form (ax + b)^n, we use the chain rule and multiply by the derivative of the inner function.

f''(x) = d/dx [f'(x)] = d/dx [2(3x + 5)^4] = 2 * 4 * (3x + 5)^(4-1) * (d/dx [3x + 5])

Simplifying, we have:
f''(x) = 8 * (3x + 5)^3 * 3 = 72 * (3x + 5)^3

Step 2: Now, we differentiate f''(x) to find f'''(x).
f'''(x) = d/dx [f''(x)] = d/dx [72 * (3x + 5)^3] = 72 * 3 * (3x + 5)^(3-1) * (d/dx [3x + 5])

Simplifying further, we have:
f'''(x) = 216 * (3x + 5)^2 * 3 = 648 * (3x + 5)^2

Step 3: We differentiate once more to find f''''(x).
f''''(x) = d/dx [f'''(x)] = d/dx [648 * (3x + 5)^2] = 648 * 2 * (3x + 5)^(2-1) * (d/dx [3x + 5])

Simplifying:
f''''(x) = 1296 * (3x + 5)

Step 4: Finally, we differentiate one more time to find the fifth derivative, f^(5)(x).
f^(5)(x) = d/dx [f''''(x)] = d/dx [1296 * (3x + 5)] = 1296 * 3 = 3888

So, the fifth derivative of f(x) at x = -5/3 is 3888.

The correct answer is option D. 3888.