Question

Differentiate `"e"^("4x" + 5)` with respect to 10^4x.

Answer 1

Let u = `"e"^(("4x" + 5))` and v = 10^4x.

u = `"e"^(("4x" + 5))`

Differentiating both sides w.r.t.x, we get

`"du"/"dx" = "e"^(("4x" + 5)) * "d"/"dx" (4"x" + 5)`

`= "e"^(("4x" + 5)) * (4 + 0)`

∴ `"du"/"dx" = 4 * "e"^(("4x" + 5)) *`

v = 10^4x 

Differentiating both sides w.r.t.x, we get

`"dv"/"dx" = 10^"4x" * log 10 * "d"/"dx" ("4x")`

∴ `"dv"/"dx" = 10^"4x" * (log 10) (4)`

∴ `"du"/"dv" = ("du"/"dx")/("dv"/"dx") = (4 * "e"^(("4x" + 5)))/(10^"4x" * (log 10)(4))`

∴ `"du"/"dv" = ("e"^(("4x" + 5)))/(10^"4x" * (log 10)`