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

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

To take the derivative, you multiply the exponent times the coefficient and then subtract one from the original exponent.

Example: 7x^2

the deriv: 14x

To find the derivative of the function f(x), you can apply the rules of differentiation. These rules include the power rule, constant rule, sum and difference rule, and product rule. Let's go step by step.

Step 1: Apply the power rule to each term.
The power rule states that if you have a term of the form ax^n, the derivative will be d/dx(ax^n) = anx^(n-1).

For the first term, 4x^(4/3), we can apply the power rule to find the derivative:
d/dx(4x^(4/3)) = 4 * (4/3)x^(4/3 - 1) = 4 * (4/3)x^(1/3) = 16/3 x^(1/3).

For the second term, (2/3)x^(3/2), we apply the power rule:
d/dx((2/3)x^(3/2)) = (2/3) * (3/2)x^(3/2 - 1) = (2/3) * (3/2)x^(1/2) = x^(1/2).

For the third term, 7x^2, we can apply the power rule:
d/dx(7x^2) = 7 * 2x^(2 - 1) = 14x.

For the fourth term, -3x, we can simply apply the constant rule since it's a linear term:
d/dx(-3x) = -3.

And the last term, 1, is a constant, so its derivative is 0.

Step 2: Combine the individual derivatives.
Now that we have found the derivatives of each term, we can combine them to get the derivative of the whole function f(x).

The derivative of f(x) is given by:

f'(x) = 16/3 x^(1/3) + x^(1/2) + 14x - 3.

Therefore, the derivative of the function f(x) is f'(x) = 16/3 x^(1/3) + x^(1/2) + 14x - 3.