Question

(b) निम्नलिखित फलनों का अवकलन कीजिए : (i) 3x +15x2 +8x+15 /3x​

Answer 1

SOLUTION

TO DETERMINE

The derivative of

$\displaystyle\sf{ \frac{3 {x}^{3} + 15 {x}^{2} + 8x + 15 }{3x} }$

EVALUATION

Let the given expression = y

Thus we get

$\displaystyle\sf{ y = \frac{3 {x}^{3} + 15 {x}^{2} + 8x + 15 }{3x} }$

$\displaystyle\sf{ \implies \: y = \frac{3 {x}^{3} }{3x} +\frac{ 15 {x}^{2} }{3x} + \frac{ 8x }{3x} + \frac{15}{3x} }$

$\displaystyle\sf{ \implies \: y = {x}^{2} + 5x + \frac{8}{3} + \frac{5}{x} }$

Differentiating both sides with respect to x we get

$\displaystyle\sf{ \implies \frac{dy}{dx} = \frac{d}{dx} \bigg( {x}^{2} + 5x + \frac{8}{3} + \frac{5}{x} \bigg) }$

$\displaystyle\sf{ \implies \frac{dy}{dx} = \frac{d}{dx} \bigg( {x}^{2} \bigg) + \frac{d}{dx} \bigg( 5x \bigg) + \frac{d}{dx} \bigg( \frac{8}{3} \bigg) + \frac{d}{dx} \bigg( \frac{5}{x} \bigg) }$

$\displaystyle\sf{ \implies \frac{dy}{dx} = 2x+ 5 + 0 - \frac{5}{ {x}^{2} } }$

$\displaystyle\sf{ \implies \frac{dy}{dx} = 2x+ 5 - \frac{5}{ {x}^{2} } }$

FINAL ANSWER

$\boxed{ \: \: \: \displaystyle\sf{ \frac{dy}{dx} = 2x+ 5 - \frac{5}{ {x}^{2} } } \: \: }$

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