Question

Differentiate y=cos^2(x^3)

Answer 1

Step-by-step explanation:

$y = { \cos}^{2}( {x}^{3} ) \\ {y}^{.} = \frac{d}{dx} ( { \cos }^{2} ({x}^{3}) ) \\ \: \: \: \: \: \: = \frac{d{ \cos }^{2} ({x}^{3}) }{d \cos( {x}^{3} ) } \times \frac{d \cos( {x}^{3})}{d {x}^{3} } \times \frac{d {x}^{3} }{dx} \\ \: \: \: \: \: \: =2\cos( {x}^{3})( - \sin( {x}^{3} ) )\times 3 {x}^{2} \\ \: \: \: \: \: \: =- 2 \cos( {x}^{3} ) \sin( {x}^{3} ) \times 3 {x}^{2} \\ \: \: \: \: \: \: = - \sin(2 {x}^{3} ) \times 3 {x}^{2}$

$y = { \cos }^{2} ( {x}^{3} ) \\ \: \: \: \: = \frac{1}{2} (2 { \cos}^{2} ( {x}^{3})) \\ \: \: \: \: = \frac{1}{2}( \cos(2 {x}^{3} ) + 1) \\ {y}^{.} = \frac{1}{2} ( \frac{d}{dx} \cos(2 {x}^{3} ) + \frac{d}{dx} 1) \\ \: \: \: \: \: = \frac{1}{2} ( - \sin(2 {x}^{3} ) \times 2 \times 3 {x}^{2} + 0) \\ \: \: \: \: = - \sin(2 {x}^{3} ) \times 3 {x}^{2}$