To find the first derivative of the function F = (1/2)S^3(2S^2 - 3S - 6), we will use the power rule and the sum/difference rule of differentiation, without using the product rule.
The given function can be rewritten as:
F = (1/2)S^3(2S^2 - 3S - 6)
F = (1/2)S^3(2S^2) - (1/2)S^3(3S) - (1/2)S^3(6)
Now, let's differentiate each term separately:
1. Differentiating the first term:
(1/2)S^3(2S^2)
Using the power rule, the derivative of (1/2)S^3 is (3/2)S^2, and the derivative of 2S^2 is 4S.
Therefore, the derivative of the first term is:
(3/2)S^2 * 2S^2 = 3S^4
2. Differentiating the second term:
-(1/2)S^3(3S)
Using the power rule, the derivative of -(1/2)S^3 is -(3/2)S^2, and the derivative of 3S is 3.
Therefore, the derivative of the second term is:
-(3/2)S^2 * 3S = -9S^3
3. Differentiating the third term:
-(1/2)S^3(6)
Using the power rule, the derivative of -(1/2)S^3 is -(3/2)S^2, and the derivative of 6 is 0.
Therefore, the derivative of the third term is:
-(3/2)S^2 * 0 = 0
Now, we can add the derivatives of the three terms to get the first derivative of the original function:
F' = 3S^4 - 9S^3 + 0
F' = 3S^4 - 9S^3
