Differentiate the following functions with respect to x(without using

1.	Differentiate the following functions with respect to x(without using first principle) :
Answer» $\huge\mathfrak{solution}$ In both questions we USE , Chain rule . $\star \boxed{ \bold{ \frac{d}{dx} f(g(x)) = f' (x) \frac{d}{dx} g(x)}}$ $1) \sqrt{sin \: x}$ Let $y = \sqrt{sin \: x}$ Differentiate w.r.t X both SIDES , $\rightarrow \: \frac{dy}{dx} = \frac{d}{dx} ( \sqrt{sin \: x} ) \\ \\ \rightarrow \: \frac{dy}{dx} = \frac{1}{2 \sqrt{sin \: x} } \frac{d}{dx} (sin \: x) \\ \\ \rightarrow \: \frac{dy}{dx} = \frac{1}{2 \sqrt{sin \: x} } (cosx) \\ \\ \rightarrow \: \frac{dy}{dx} = \frac{cos \: x}{2 \sqrt{sin \: x} }$ $2) \: sin \: \sqrt{x}$ Let $y = sin \: \sqrt{x}$ Differentiate w.r.t X both sides, $\star \frac{dy}{dx} = \frac{d}{dx} (sin \: \sqrt{x} ) \\ \\ \star \frac{dy}{dx} = cos \: \sqrt{x} \frac{d}{dx} ( \sqrt{x} ) \\ \\ \star \frac{dy}{dx} = cos \: \sqrt{x} ( \frac{1}{2 \sqrt{x} } ) \\ \\ \star \: \frac{dy}{dx} = \frac{cos \: \sqrt{x} }{2 \sqrt{x} }$ Formula USED: $\red \star \boxed{ \bold{ \frac{d}{dx} ( \sqrt{x} ) = \frac{1}{2 \sqrt{x} } }}$ $\green\star \boxed{\bold{ \frac{d}{dx} (sin \: x) = cos \: x}}$

Differentiate the following functions with respect to x(without using first principle) :

Answer»

$\huge\mathfrak{solution}$

In both questions we USE ,

Chain rule .

$\star \boxed{ \bold{ \frac{d}{dx} f(g(x)) = f' (x) \frac{d}{dx} g(x)}}$

$1) \sqrt{sin \: x}$

Let

$y = \sqrt{sin \: x}$

Differentiate w.r.t X both SIDES ,

$\rightarrow \: \frac{dy}{dx} = \frac{d}{dx} ( \sqrt{sin \: x} ) \\ \\ \rightarrow \: \frac{dy}{dx} = \frac{1}{2 \sqrt{sin \: x} } \frac{d}{dx} (sin \: x) \\ \\ \rightarrow \: \frac{dy}{dx} = \frac{1}{2 \sqrt{sin \: x} } (cosx) \\ \\ \rightarrow \: \frac{dy}{dx} = \frac{cos \: x}{2 \sqrt{sin \: x} }$

$2) \: sin \: \sqrt{x}$

Let

$y = sin \: \sqrt{x}$

Differentiate w.r.t X both sides,

$\star \frac{dy}{dx} = \frac{d}{dx} (sin \: \sqrt{x} ) \\ \\ \star \frac{dy}{dx} = cos \: \sqrt{x} \frac{d}{dx} ( \sqrt{x} ) \\ \\ \star \frac{dy}{dx} = cos \: \sqrt{x} ( \frac{1}{2 \sqrt{x} } ) \\ \\ \star \: \frac{dy}{dx} = \frac{cos \: \sqrt{x} }{2 \sqrt{x} }$

Formula USED:

$\red \star \boxed{ \bold{ \frac{d}{dx} ( \sqrt{x} ) = \frac{1}{2 \sqrt{x} } }}$

$\green\star \boxed{\bold{ \frac{d}{dx} (sin \: x) = cos \: x}}$