Differentiate, with respect to x, \(x^{3} + 2x\) from the first principle.

Explanation

\(y = x^{3} + 2x ... (1)\)

Let an increment in x = \(\Delta x\) and an increment in y = \(\Delta y\).

Then, \(y + \Delta y = (x + \Delta x)^{3} + 2(x + \Delta x)\)

\(y + \Delta y = x^{3} + 3x^{2} \Delta x + 3x (\Delta x)^{2} + (\Delta x)^{3} + 2x + 2 \Delta x ... (2)\)

\((2) - (1) : \Delta y = 3x^{2} \Delta x + 3x (\Delta x)^{2} + (\Delta x)^{3} + 2 \Delta x\)

\(\frac{\Delta y}{\Delta x} = 3x^{2} + 3x \Delta x + (\Delta x)^{2} + 2\)

\(\frac{\mathrm d y}{\mathrm d x} = \lim \limits_ {\Delta x \to 0} \frac{\Delta y}{\Delta x}\)

\(\frac{\mathrm d y}{\mathrm d x} = \lim \limits_{\Delta x \to 0} (3x^{2} + 3x \Delta x + (\Delta x)^{2} + 2 = 3x^{2} + 2\)