Question

Find the differential \(d y\). \(y=3 x^{2}-4\)

Short answer

The differential \(d y\) of the function \(y = 3x^{2} - 4\) is \(dy = 6x\).

Step-by-step solution

Identify the function

The function given in this problem is \(y = 3x^{2} - 4\). We are asked find the differential \((dy)\) of the function.

Understand the definition

The differential of a function, noted as \(dy\), is equal to the derivative of the function (\(y'\)) with respect to \(x\) times \(dx\). This means, if \(y = f(x)\), then \(dy = f'(x) \cdot dx\). In this exercise, \(dx\) is understood to be 1, so differential \(dy\) will be equivalent to derivative \(f'(x)\).

Differentiate the function

Differentiate the function \(y = 3x^{2} - 4\) with respect to \(x\). To do so, apply the power rule for differentiation, which states that the derivative of \(x^n\) is \(n \cdot x^{n-1}\). So for the first term, \(3x^{2}\), the derivative is \(2 \cdot 3x^{2-1}\), or \(6x\). Since the derivative of a constant is 0, the derivative of \(-4\) is 0. Therefore, \(dy = 6x - 0 = 6x\).