Question

x के साक्षेप निम्नलिखित फलन का अवकलन कीजिए:

${\displaystyle {\cos x}^3.{\sin }^2\left(x^5\right)}$

Answer 1

${\displaystyle {\cos x}^3.{\sin }^2\left(x^5\right)}$

मान लीजिए y = ${\displaystyle {\cos x}^3.{\sin }^2\left(x^5\right)}$

${\displaystyle \frac{dy}{dx}=\frac{d\left({\cos x}^3.{\sin }^2\left(x^5\right)\right)}{dx}}$ भागफल नियम द्वारा,

${\displaystyle =\frac{{\cos x}^3 d{\sin }^2\left(x^5\right)}{dx}+\frac{{\sin }^2\left(x^5\right)d{\cos x}^3}{dx}}$ श्रृंखला नियम द्वारा,

${\displaystyle =\frac{{\cos x}^3\left(2\sin \left(x^5\right)\right).d\sin \left(x^5\right)}{dx}+\frac{{\sin }^2\left(x^5\right)(-{\sin x}^3).d x^3}{dx}}$

${\displaystyle =\frac{{\cos x}^3\left(2\sin \left(x^5\right)\right).{\cos x}^5{dx}^5}{dx}+{\sin }^2\left(x^5\right)(-{\sin x}^3)3x^2}$ 

${\displaystyle ={\cos x}^{3}2\sin \left(x^5\right){\cos x}^5.5x^4+{\sin }^2\left(x^5\right)(-{\sin x}^3)3x^2}$

${\displaystyle =10 x^4{\sin x}^5{\cos x}^5{\cos x}^3-{\sin x}^3{\sin }^2{\sin }^2{x}^5}$