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)