Question

Find the differential \(d y\). \(y=\sqrt[3]{6 x^{2}}\)

Short answer

The differential of the given function \(y=\sqrt[3]{6 x^{2}}\) is \(dy=4x(6x^2)^{-2/3}dx\).

Step-by-step solution

Rewrite the function in a more convenient form

First, rewrite the function in a form that makes applying the power rule of differentiation easier. Rewrite the cube root as a power of 1/3: \(y=(6x^2)^{1/3}\).

Apply the Chain Rule

The Chain Rule is as follows: If \(f(x)\) and \(g(x)\) are functions, then the derivative of the composite function \(f(g(x))\) is \(f'(g(x)) * g'(x)\). In our case, let's take \(u=6x^2\) and \(f(x)=x^{1/3}\). Our integral function then becomes a composite function: \(y=f(u)\). Differentiate \(u\) with respect to \(x\) (i.e., calculate \(du/dx\)) and differentiate \(f(u)\) with respect to \(u\) (i.e., calculate \(df/du\)). For \(f'(u)\), we get \(1/3u^{-2/3}\), and for \(du/dx\), we get \(12x\).

Assemble the differential

We can now calculate \(dy/dx\) using the chain rule \(dy/dx=f'(u)*du/dx\). Substituting the differentials we just calculated, we get \(dy/dx=(1/3u^{-2/3})*(12x)\). Substituting back the original \(u=6x^2\) we get \(dy/dx=(1/3*(6x^2)^{-2/3})*(12x)\), which simplifies to \(dy/dx=4x(6x^2)^{-2/3}\). Now, the differential \(dy\) is simply \(dy/dx\) multiplied by \(dx\), so \(dy=4x(6x^2)^{-2/3}dx\).