Question

differentiate sin x ^ 2 w.rt. x ^ 3​

Answer 1

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▪ Let :-

$\sf { \sin \mathtt x}^{2} = A \\ \bf and \\ \sf {\mathtt x}^{3} = B$

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▪ To Find :-

$\sf Derivative \: of \: \sin\mathtt x {}^{2} \: w.r.t. \: \mathtt x \\ \\ \sf Which \: \: Means, \\ \\ \bold{ We \: have \: to \: find} \: \: \red{\pmb{ \dfrac{dA}{dB} }}$

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▪ Solution :-

We Have,

$\sf A = { \sin \mathtt x}^{2}$

$\large \bigstar \: \underline{ \pmb{ \mathfrak{ Differentiating \: both \: sides \: \bf w.r.t \: x }}}$

$:\longmapsto \sf \dfrac{dA}{d \mathtt x} = \frac{d}{d \mathtt x} ( \sin { \mathtt x}^{2} ) \\ \\ = \sf \cos { \mathtt x}^{2} . \frac{d}{d \mathtt x} ( { \mathtt x}^{2} ) \\ \\ \large :\longmapsto \pmb{ \boxed{\frac{dA}{d x} = 2x. \cos x {}^{2} }}$

Now We Also Have ,

$\sf B = { \mathtt x}^{3}$

$\large \bigstar \: \underline{ \pmb{ \mathfrak{ Differentiating \: both \: sides \: \bf w.r.t \: x }}}$

$:\longmapsto \sf \dfrac{dB}{d \mathtt x} = \dfrac{d}{d \mathtt x} ( { \mathtt x}^{3} ) \\ \\ \large : \longmapsto \pmb{ \boxed{ \dfrac{dB}{dx} = 3 {x}^{2} }}$

Now,

$\sf \dfrac{dA}{dB} = \dfrac{dA/dx}{dB/dx} \\ \\ = \sf \frac{2 \: \cancel\mathtt x \: . \: \cos { \mathtt x}^{2} }{3 { \mathtt x}^{ \cancel2} } \\ \\\Large \purple{ :\longmapsto  \underline { \pmb{\boxed{{ \frac{dA}{dB} = \frac{2 \cos {x}^{2} }{3x} } }}}}$

$\Large \red{\mathfrak{  \text{W}hich \:   \: is  \:  \: the  \:  \: required} }\\ \huge \red{\mathfrak{ \text{ A}nswer.}}$

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