Question

Differentiate the following with regard to x : y=(cosec x +2tanx)^3

Answer 1

$y' = {\left( {{{\left( {\cos ecx + 2\tan x} \right)}^3}} \right)^\prime } = \\=3{\left( {\cos ecx + 2\tan x} \right)^2} \cdot {\left( {\cos ecx + 2\tan x} \right)^\prime } = 3{\left( {\cos ecx + 2\tan x} \right)^2} \cdot {\left( {\frac{1}{{\sin x}} + 2\tan x} \right)^\prime } =\\ =3{\left( {\cos ecx + 2\tan x} \right)^2} \cdot \left( { - \frac{1}{{{{\sin }^2}x}}\cos x + \frac{2}{{{{\cos }^2}x}}} \right) = 3{\left( {\cos ecx + 2\tan x} \right)^2} \cdot \left( { - \frac{{cotx}}{{\sin x}} + \frac{2}{{{{\cos }^2}x}}} \right) =\\= 3{\left( {\cos ecx + 2\tan x} \right)^2} \cdot \left( {2{{\sec }^2}x - cotx\cos ecx} \right)$