Math, asked by PalakKharbanda, 1 month ago

find dy/dx of y=x^tanx + √x+1/x​

Answers

Answered by sandy1816
2

y =  {x}^{tanx}  +  \sqrt{x +  \frac{1}{x} }  \\ let \:  \:  \: u =  {x}^{tanx}  \:  \:  \:  \: v =  \sqrt{x +  \frac{1}{x} }  \\  so \:  \:  \:  \: y = u + v \\  \frac{dy}{dx}  =  \frac{du}{dx}  +  \frac{dv}{dx} ...(1) \\ \\  now \:  \:  \:  \: u =  {x}^{tanx} \\ applying \: log \: both \:  \: sides  \\ logu = tanxlogx \\ diff. \:  \:  \: w.r.t \:  \:  \: x \\  \frac{1}{u}  \frac{du}{dx}  = tanx \frac{1}{x}  + logx( {sec}^{2} x) \\  \frac{du}{dx}  =  {x}^{tanx} ( \frac{tanx}{x}  +  {sec}^{2} x \: logx) \\  \\ now \:  \:  \: v =  \sqrt{x +  \frac{1}{x} }  \\  \frac{dv}{dx}  =  \frac{1}{2 \sqrt{x +  \frac{1}{x} } } (1 -  \frac{1}{ {x}^{2} } ) \\  \\ from \:  \:  \: (1) \\  \frac{dy}{dx}  =  {x}^{tanx} ( \frac{tanx}{x}  +  {sec}^{2} x \: logx) +  \frac{(1 -  \frac{1}{ {x}^{2} }) }{2 \sqrt{x +  \frac{1}{x} } }  \\

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