Question

Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:

`sqrt(2x + 5)` at x = 2

Answer 1

Let f(x) = `sqrt(2x + 5)` 

∴ f(2) = `sqrt(2(2) + 5) = sqrt(9)` = 3 and

f(2 + h) = `sqrt(2(2 + "h") + 5) = sqrt(2"h" + 9)`

By first principle, we get

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

∴ f'(2) = `lim_("h" -> 0) ("f"(2 + "h") - "f"(2))/"h"`

= `lim_("h" -> 0) (sqrt(2"h" + 9) - 3)/"h"`

= `lim_("h" -> 0) (sqrt(2"h" + 9) - 3)/"h" xx (sqrt(2"h" + 9) + 3)/(sqrt(2"h" + 9) + 3)`

= `lim_("h" -> 0) (2"h" + 9 - 9)/("h"(sqrt(2"h" + 9) + 3)`

= `lim_("h" -> 0) (2"h")/("h"(sqrt(2"h" + 9) + 3)`

= `lim_("h" -> 0) 2/(sqrt(2"h" + 9) + 3)`  ...[∵ h → 0, ∴ h ≠ 0]

= `2/(sqrt(0 + 9) + 3)`

= `2/(3 + 3)`

= `1/3`