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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
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Solution
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`
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