Question

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)