Find the derivative of the function f by using the rules of differentiation.

f(x)=7x^4/3-2/3x^3/2+2x^2-5x+1

f '(x) = ??

without brackets, I cannot tell if you meant

7x^(4/3) or (7x^4)/3
Brackets in this case are essential

there were no brackets in this problem

To find the derivative of a function, f(x), using the rules of differentiation, we can differentiate each term of the function separately. Let's find the derivative of each term:

1. The derivative of 7x^(4/3):
To differentiate x^(4/3), we use the power rule. The power rule states that if we have a term of the form x^n, the derivative is n*x^(n-1). Applying this rule, the derivative of x^(4/3) is (4/3) * x^(4/3 - 1) = (4/3) * x^(1/3). Therefore, the derivative of 7x^(4/3) is 7 * (4/3) * x^(1/3) = (28/3) * x^(1/3).

2. The derivative of -2/3x^(3/2):
Similarly, using the power rule, the derivative of x^(3/2) is (3/2) * x^(3/2 - 1) = (3/2) * x^(1/2). Therefore, the derivative of -2/3x^(3/2) is -2/3 * (3/2) * x^(1/2) = -x^(1/2).

3. The derivative of 2x^2:
Using the power rule, the derivative of x^2 is 2 * x^(2-1) = 2 * x. Therefore, the derivative of 2x^2 is 2 * 2 * x = 4x.

4. The derivative of -5x:
The derivative of -5x is simply -5 since the derivative of x is 1.

5. The derivative of 1:
The derivative of a constant is always zero since it does not depend on the variable x.

Now, summing up all the derivatives of each term, we have:

f '(x) = (28/3) * x^(1/3) - x^(1/2) + 4x - 5

Therefore, the derivative of f(x) is f '(x) = (28/3) * x^(1/3) - x^(1/2) + 4x - 5.