Question

Find the derivative of the following function by the first principle: 3x² + 4

Answer 1

Let f(x) = 3x² + 4
∴ f(x + h) = 3(x + h)² + 4
= 3(x² + 2xh + h²) + 4
= 3x² + 6xh + 3h² + 4
By first principle, we get

f ′(x) =`lim_("h"→ 0) ("f"(x + "h") - "f"(x))/"h"`

=`lim_("h" → 0) ((3x^2 + 6x"h" + 3"h"^2 + 4) - (3x^2 + 4))/"h"`

=`lim_("h" → 0) (3"h"^2 + 6x"h")/"h"`

=`lim_("h"→0) (h(3h+6x))/h`

=`lim_("h" → 0)(6x + 3"h")`   …[∵ h → 0, ∴h ≠ 0]

= 6x + 3(0)
= 6x