Question

DIfferentiate x sin x w.r.t. tan x.

Answer 1

Let u = x sin x and v = tan x.
Then we want to find `"du"/"dv"`
Differentiating u and v w.r.t. x, we get
`"du"/"dx" = "d"/"dx"(x sin x)`

= `x"d"/"dx"(sin x) + (sin x)."d"/"dx"(x)`
= x cos x + (sin x) x 1
= x cos x + sin x
and
`"dv"/"dx" = "d"/"dx"(tanx)` = sec²x

∴ `"du"/"dv" = (("du"/"dx"))/(("dv"/"dx")`

= `(x cos x + sin x)/(sec^2x)`.