basic calculus

Calculus Level 2

Given that y = ( sin ⁡ x ) 2 y= (\sin x)^2 , find d y d x \dfrac {dy}{dx} .

sin ⁡ x \sin x sin ⁡ 4 x \sin 4x cos ⁡ x \cos x cos ⁡ 4 x \cos 4x cos ⁡ 2 x \cos 2x sin ⁡ 2 x \sin 2x

Mohammad Khaza by Mohammad Khaza , 20, Bangladesh

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2 solutions

Munem Shahriar
Oct 5, 2017

Relevant wiki: Chain Rule

d ( sin ⁡ x ) 2 d x \dfrac{d(\sin x)^2}{dx}

= d d x ( sin ⁡ x ) 2 = \dfrac{d}{dx} (\sin x)^2

= d d u ( u 2 ) d d x sin ⁡ ( x ) = \dfrac{d}{du} (u^2) \dfrac{d}{dx} \sin(x)

= 2 u cos ⁡ ( x ) = 2u \cos (x)

= 2 sin ⁡ ( x ) cos ⁡ ( x ) = 2 \sin (x) \cos(x)

= sin ⁡ 2 x = \boxed{\sin 2x}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

that is nicely explained.

Mohammad Khaza - 3 years, 8 months ago
Mohammad Khaza
Oct 4, 2017

suppose,sinx=Z, and \text{suppose,sinx=Z, and} y = Z 2 y=Z^2

so now, d y d x \frac{dy}{dx} = d y d z . d z d x \frac{dy}{dz} . \frac{dz}{dx} = 2 z . c o s x 2z . cosx = 2 s i n x c o s x 2 sinx cosx = s i n 2 x sin2x

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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