Question

Find the differential \(d y\). \(y=(4 x-1)^{3}\)

Short answer

The differential \(dy\) is \(dy = 12(4x - 1)^{2}\)

Step-by-step solution

Identify the Inner and Outer Function

In the given function \(y=(4x - 1)^{3}\), the inside function is \(u = 4x - 1\), and the outside function is \(v = u^{3}\).

Differentiate the Inner and Outer Function

Differentiating the inside function \(u=4x - 1\), we get \(du/dx = 4\). Differentiating the outside function \(v = u^{3}\), we get \(dv/du = 3u^{2}\).

Apply the Chain Rule

The chain rule states that \(dy/dx = dv/du * du/dx\). Substituting the derivatives calculated in step 2, this gives us \(dy/dx = 3*(4x - 1)^{2}*4\).