1.

Differentiate the following functions with respect to x (without using first principle) :i) Sin 2xii) Sin² xiii) Sin x²​

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\huge \red{ \mathfrak{solution}}

We USE chain RULE in all questions :

\star \bold{ \frac{d}{dx} f(g(<klux>X</klux>)) = f' (x) \frac{d}{dx} g(x)}

1) \: sin2x

LET

y = \: sin \: 2x

Differentiate w.r.t X both sides ,

\star \frac{dy}{dx} = \frac{d}{dx} (sin \: 2x) \\ \\ \star \: \frac{dy}{dx} = cos \: 2x \frac{d}{dx} (2x) \\ \\ \star \: \frac{dy}{dx} = cos \: 2x(2) \\ \\ \star \: \frac{dy}{dx} = 2 \: cos \: 2x

Formula:

\star \bold{ \frac{dy}{dx} sinx = cosx}

2) y = {sin}^{2} x

Differentiate w.r.t X both sides,

\star \: \frac{dy}{dx} = \frac{d}{dx} ( {sin}^{2} x) \\ \\ \star \frac{dy}{dx} = 2 \: sinx \frac{d}{dx} (sin \: x) \\ \\ \star \: \frac{dy}{dx} = 2 \: sinx \: cosx \\ \\ \star \: \frac{dy}{dx} = sin2x

Formula :

\leadsto \boxed{ \bold{ \frac{d}{dx} ( {x}^{n} ) = n {x}^{n - 1} }}

\leadsto \boxed { \bold {sin \: 2x = 2 \: sinx \: cosx}}

3) \: y = sin \: {x}^{2}

Differentiate w.r.t.X both sides ,

\star \: \frac{dy}{dx} = \frac{d}{dx} sin {x}^{2} \\ \\ \star \frac{dy}{dx} = cos \: {x}^{2} \frac{d}{dx} {x}^{2} \\ \\ \star \: \frac{dy}{dx} = cos \: {x}^{2} (2x) \\ \\ \star \: \frac{dy}{dx} = 2x \: cos \: {x}^{2}


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