Question

Find the derivative of the following w. r. t. x by using method of first principle:

x² + 3x – 1

Answer 1

Let f(x) = x² + 3x – 1

∴ f(x + h) = (x + h)² + 3(x + h) – 1

= x² + 2xh + h² + 3x + 3h – 1

By first principle, we get

f'(x) = `lim_("h" -> 0) ("f"(x + "h") - "f"(x))/"h"`

= `lim_("h" -> 0) ([x^2 + 2x"h" + "h"^2 + 3x + 3"h" - 1] - [x^2 + 3x - 1])/"h"`

= `lim_("h" -> 0) (2x"h" + "h"^2 + 3"h")/"h"`

= `lim_("h" -> 0) ("h"(2x + "h" + 3))/"h"`

= `lim_("h" -> 0) (2x + "h" + 3)`   ...[∵ h → 0, ∴ h ≠ 0]

= 2x + 3