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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:
tan x at x = `pi/4`
Solution
Let f(x) = tan x
∴ `"f"(pi/4) = tan pi/4`
`"f"(pi/4 + "h") = tan (pi/4 + "h")`
By definition,
`"f'"(pi/4) = lim_("h" -> 0) ("f"(pi/4 + "h") - "f"(pi/4))/"h"`
= `lim_("h" -> 0) (tan(pi/4 + "h") - tan pi/4)/"h"`
= `lim_("h" -> 0) (tan[(pi/4 + "h") - pi/4][1 + tan(pi/4 + "h") tan pi/4])/"h" ...[because tan("A" - "B") = (tan "A" - tan "B")/(1 + tan "A" tan "B")]`
= `lim_("h" -> 0) (tan"h"[1 + tan(pi/4 + "h") tan pi/4])/"h"`
= `[lim_("h" -> 0) tan"h"/"h"] xx [lim_("h" -> 0) {1 + tan(pi/4 + "h") tan pi/4}]`
= `1 xx [1 + tan(pi/4 + 0) tan pi/4]`
= 2 ...`[because tan pi/4 = 1]`
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