Question

$\frac{d^4 y}{dx^4} (3x^4) = \square$

Explanation:

Step1: Recall the formula for nth derivative of \(x^n\)

The nth derivative of \(x^n\) with respect to \(x\) is given by \(n!\), and for \(x^k\) where \(k < n\), the nth derivative is \(0\). Here we need to find the fourth derivative of \(3x^4\) with respect to \(x\), so first find the fourth derivative of \(x^4\).
The first derivative of \(x^4\) is \(\frac{d}{dx}(x^4)=4x^3\)
The second derivative is \(\frac{d^2}{dx^2}(x^4)=\frac{d}{dx}(4x^3) = 12x^2\)
The third derivative is \(\frac{d^3}{dx^3}(x^4)=\frac{d}{dx}(12x^2)=24x\)
The fourth derivative is \(\frac{d^4}{dx^4}(x^4)=\frac{d}{dx}(24x) = 24\)

Step2: Multiply by the constant factor

We have the function \(y = 3x^4\), so \(\frac{d^4y}{dx^4}=\frac{d^4}{dx^4}(3x^4)=3\times\frac{d^4}{dx^4}(x^4)\)
From step 1, \(\frac{d^4}{dx^4}(x^4) = 24\), so \(3\times24 = 72\)

Answer:

\(72\)