find the derivative of the given equation f(x) = x³ + x² + 3

Explanation:

Step1: Apply the power rule

The power rule states that if ( y = x^n ), then ( y^\prime=nx^{n - 1} ). For ( y=x^{3}), (y^\prime = 3x^{3-1}=3x^{2}). For (y = x^{2}), (y^\prime=2x^{2 - 1}=2x). For (y = 3=3x^{0}), (y^\prime=3\times0x^{-1}=0).

Step2: Sum the derivatives

Since (f(x)=x^{3}+x^{2}+3), by the sum rule ((u + v+w)^\prime=u^\prime + v^\prime+w^\prime). Here (u = x^{3}), (v=x^{2}), (w = 3). So (f^\prime(x)=(x^{3})^\prime+(x^{2})^\prime+(3)^\prime). Substituting the derivatives from step 1: (f^\prime(x)=3x^{2}+2x+0).

Answer:

(3x^{2}+2x) (the third option)