Question

Find the differential \(d y\). \(y=3 x^{2 / 3}\)

Short answer

The differential \(dy\) is given by \(dy = 2x^{-1/3} dx\).

Step-by-step solution

Understanding the Power Rule

The power rule for derivatives can be stated as follows: if \(y = x^n\), then the derivative \(\(dy/dx\) = n*x^{n-1}\). In this case, \(y = 3x^{2/3}\), so the base \(x\) is raised to the power of \(2/3\).

Apply the Power Rule

Applying the power rule to the function, differentiate it with respect to \(x\). That gives us \(dy/dx = (2/3) * 3x^{(2/3) - 1}\). Simplify this expression to make the calculation more precise.

Simplify the Expression

Simplifying the expression gives \(dy/dx = 2x^{-1/3}\).

Determine the Differential dy

The differential dy is found by multiplying \(dy/dx\) by \(dx\). So \(dy = 2x^{-1/3} dx\).