Question

find the derivative of the given equation f(x)=x^4 + 4x^3 - 2x^2

Explanation:

Step1: Apply the power rule

The power rule states that if \(y = x^n\), then \(y^\prime=nx^{n - 1}\).
For \(y=x^{4}\), \(y^\prime = 4x^{4-1}=4x^{3}\).
For \(y = 4x^{3}\), \(y^\prime=4\times3x^{3 - 1}=12x^{2}\).
For \(y=-2x^{2}\), \(y^\prime=-2\times2x^{2-1}=-4x\).

Step2: Sum up the derivatives

Since \(f(x)=x^{4}+4x^{3}-2x^{2}\), by the sum - rule \((u + v+w)^\prime=u^\prime + v^\prime+w^\prime\) (where \(u = x^{4}\), \(v = 4x^{3}\), \(w=-2x^{2}\)).
\(f^\prime(x)=(x^{4})^\prime+(4x^{3})^\prime+(-2x^{2})^\prime\)
\(f^\prime(x)=4x^{3}+12x^{2}-4x\)

Answer:

\(4x^{3}+12x^{2}-4x\) (the fourth option)