Question

differentiate the following w.r.t x (1+sin^2x)^2 + (1+cos^2x)^3
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Answer 1

Step-by-step explanation:

let us suppose,

$f(x) = {(1 + { \sin}^{2}(x) )}^{2} + {(1 + { \cos}^{2} (x) )}^{3} \\ \\ f'(x) = \frac{d(f(x))}{dx} =(2) (1 + { \sin }^{2} (x))(2)( \cos(x) ) + 3({1 + { \cos}^{2} (x) )}^{2} (2)( - \sin(x) ) \\ \\ simplyfying \: \: this.... \\ \\ f'(x) = 4 \cos(x) + 4 { \sin}^{2} (x) \cos(x) - 6 \sin(x) {(1 + { \cos }^{2}(x) )}^{2} \\ \\ f'(x) = 4 { \sin }^{3} (x) \cos(x) - \sin(2x) - 12 { \cos}^{3} (x) \sin(x) - 6 { \cos }^{5} (x) \sin(x)$

Answer 2

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y = (1 + sin²(x))² + (1 + cos² (x)) ³

So; The first derivative can be given as:

= (1 + sin²x)² + (1 + cos²x)³

Let;

n = 1 + sin²x and u = 1 + cos²x

d dn (n)². d (1 + sin²x) + du (u)³. (1+ cos²x)

dy2(1+ sin²x). (2) (cos(x)) + 3(1 + cos²x)2(2)(-sina)

dy 4cosx+4sin2x.cosæ 6sinx.(1 + cos²x) ²

dy 4 sin ³(x) cos(x) = sin(2x) - = 12cos³ (x) sin(x) - 6 cos ³ (x) sin(x)

• I Hope it's Helpful My Friend.