Question

Differentiate y wrt x , y=( 1+sin^2x)^2 + (1+cos^2x)^3

Answer 1

$y = (1 + \sin ^{2} ( {x} ) ) ^{2} + ( {1 + { \cos ^{2} (x) } )}^{3}$

So; The first derivative can be given as:

$\frac{dy}{dx} = \frac{d}{dx} (1 + sin ^{2} x) ^{2} + \frac{d}{dx} (1 + {cos}^{2} x) ^{ 3}$

Let; n = 1 + sin²x and u = 1 + cos²x

$\frac{dy}{dx} = \frac{d}{dn} (n) ^{2} . \frac{d}{dx} (1 + {sin}^{2} x) + \frac{d}{du} (u) ^{3} . \frac{d}{dx} (1 + {cos}^{2} x)$

$\frac{dy}{dx} = 2(1 + sin {}^{2} x).(2)( \cos(x) ) + 3(1 + cos ^{2} x) {}^{2} (2)( - sinx)$

$\frac{dy}{dx} = 4cosx + 4 \sin {}^{2} x. \: cosx - 6 sinx.(1 + cos {}^{2} x) {}^{2}$

$\frac{dy}{dx} = 4 \sin {}^{3} (x) \cos(x) - \sin(2x) - 12cos {}^{3} (x) \sin(x) - 6 \cos {}^{5} (x) \sin(x)$