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`

Answer 1

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]`