Respuesta :
Answer:
Given that:
[tex]\longmapsto{ \large{ \rm{y = \sin \:x \degree}}}[/tex]
We know that,
[tex] \dashrightarrow{ \boxed{ \red{ \rm{1 \degree = \left ( \frac{\pi}{100} \right)^{c} }}}}[/tex]
So, using this, above given can be written as,
[tex]{ \large{ \longrightarrow{ \rm{y = \sin \left( \frac{\pi x}{180} \right) }}}}[/tex]
On differentiating both sides w.r.t. x, we get:
[tex]{ \large { \longrightarrow{ \rm{ \frac{d}{dx}y = \frac{d}{dx} \sin \left( \frac{\pi x}{180} \right) }}}}[/tex]
We know that,
[tex]{ \dashrightarrow{ \boxed{ \red{ \rm{ \frac{d}{dx} \sin x = \cos x }}}}}[/tex]
So, using the result, we get:
[tex]{ \large{ \longrightarrow{ \rm{ \frac{d}{dx} = \cos \left( \frac{\pi \: x}{180} \: \right) \frac{d}{dx} \left( \frac{\pi \: x}{180} \right) }}}}[/tex]
We know that,
[tex]{ \dashrightarrow{ \boxed{ \red{ \rm{ \frac{d}{dx}k \: f(x) = k \frac{d}{dx} \: f(x)}}}}}[/tex]
So, using this, we get:
[tex]{ \large{ \longrightarrow{ \rm{ \frac{dy}{dx} = \cos \left( \frac{\pi \: x}{180} \right) \: \times \: \left( \frac{\pi}{180} \right) \frac{d}{dx} x}}}}[/tex]
We know that,
[tex]{ \dashrightarrow{ \boxed{ \red{ \rm{ \frac{d}{dx} x= 1 }}}}}[/tex]
So, using this, we get:
[tex]{ \large{ \longrightarrow{ \rm{ \frac{dy}{dx} = \left( \frac{\pi}{180} \right) \: \cos \left( \frac{\pi \: x}{180} \right ) \times 1}}}}[/tex]
Hence:
[tex]{ \large{ \leadsto{ \green{ \rm{ \frac{dy}{dx} = \left( \frac{\pi}{180} \right) \cos \left( \frac{\pi \: x}{180} \right) }}}}}[/tex]
OR
[tex]{ \large{ \leadsto{ \green{ \rm{ \frac{dy}{dx} = \left( \frac{\pi}{180} \right) \cos \: x \degree }}}}}[/tex]
[tex] \: \: [/tex]
Learn More:
[tex] \boxed{\begin{array}{c|c}\bf f(x)&\bf\dfrac{d}{dx}f(x)\\ \\ \frac{\qquad\qquad}{}&\frac{\qquad\qquad}{}\\ \sf k&\sf0\\ \\ \sf sin(x)&\sf cos(x)\\ \\ \sf cos(x)&\sf-sin(x)\\ \\ \sf tan(x)&\sf{sec}^{2}(x)\\ \\ \sf cot(x)&\sf-{cosec}^{2}(x)\\ \\ \sf sec(x)&\sf sec(x)tan(x)\\ \\ \sf cosec(x)&\sf-cosec(x)cot(x)\\ \\ \sf\sqrt{x}&\sf\dfrac{1}{2\sqrt{x}}\\ \\ \sf log(x)&\sf\dfrac{1}{x}\\ \\ \sf{e}^{x}&\sf{e}^{x}\end{array}}[/tex]
