Question

find dy.
\\( y = 26 + 8x^3 - 4x^4 \\)
\\( dy = \square \\) (simplify your answer.)

Explanation:

Step1: Differentiate each term

Differentiate \(26\) (derivative of constant is \(0\)), \(8x^{3}\) (use power rule \((x^n)'=nx^{n - 1}\)), and \(-4x^{4}\) (use power rule).
For \(y = 26+8x^{3}-4x^{4}\), we have \(\frac{dy}{dx}=\frac{d}{dx}(26)+\frac{d}{dx}(8x^{3})-\frac{d}{dx}(4x^{4})\).
Since \(\frac{d}{dx}(26) = 0\), \(\frac{d}{dx}(8x^{3})=8\times3x^{2}=24x^{2}\), \(\frac{d}{dx}(4x^{4})=4\times4x^{3}=16x^{3}\).
So \(\frac{dy}{dx}=24x^{2}-16x^{3}\).

Step2: Find \(dy\)

Recall that \(dy=\frac{dy}{dx}dx\).
Since \(\frac{dy}{dx}=24x^{2}-16x^{3}\), then \(dy=(24x^{2}-16x^{3})dx\).

Answer:

\((24x^{2}-16x^{3})dx\)