Question

Plz solve if possible.​

Answer 1

$\rm \longrightarrow y = \dfrac{ log(x) }{x} + {e}^{x} sin \: x + { log_{5}(x) } \\ \\ \\ \rm \longrightarrow y = \dfrac{ log(x) }{x} + {e}^{x} sin \: x + \dfrac{ log(x) }{log(5)} \\ \\ \\ \rm \longrightarrow \dfrac{dy}{dx} = \frac{d \bigg(\dfrac{ log(x) }{x} + {e}^{x} sin \: x + \dfrac{ log(x) }{log(5)} \bigg )}{dx}$

$\rm \longrightarrow \dfrac{dy}{dx} = \dfrac{d( \dfrac{log \: x}{x}) }{dx} + \dfrac{d( {e}^{x} \: sin \: x)}{dx} + \dfrac{1}{log \: 5} \dfrac{d(log \: x)}{dx} \\ \\ \\ \rm \longrightarrow \dfrac{dy}{dx} = \frac{x \times \frac{1}{x} - log( x) }{ {x}^{2} } + {e}^{x} \: sin \: x + {e}^{x} cos \: x+ ( \dfrac{1}{x} \times\dfrac{1} {log \: 5} ) \\ \\ \\ \rm \longrightarrow \dfrac{dy}{dx} = \frac{1- log( x) }{ {x}^{2} } + {e}^{x}( \: sin \: x + cos \: x)+ \dfrac{1} {x \: log \: 5}$

Answer :

$\rm \dfrac{dy}{dx} = \frac{1- log( x) }{ {x}^{2} } + {e}^{x}( \: sin \: x + cos \: x)+ \dfrac{1} {x \: log \: 5}$

_________________

d(e^x)/dx = e^x

d(x^n)/dx = n x^(n-1)

d(ln x)/dx = 1/x

d(sin x)/dx = cos x

d(cos x)/dx = - sin x

d(tan x)/dx = sec² x

d(sec x)/dx = tan x * sec x

d(cot x)/dx = - cosec²x

d(cosec x)/dx = - cosec x * cot x

Answer 2

\begin{gathered} \rm \longrightarrow y = \dfrac{ log(x) }{x} + {e}^{x} sin \: x + { log_{5}(x) } \\ \\ \\ \rm \longrightarrow y = \dfrac{ log(x) }{x} + {e}^{x} sin \: x + \dfrac{ log(x) }{log(5)} \\ \\ \\ \rm \longrightarrow \dfrac{dy}{dx} = \frac{d \bigg(\dfrac{ log(x) }{x} + {e}^{x} sin \: x + \dfrac{ log(x) }{log(5)} \bigg )}{dx} \end{gathered}⟶y=xlog(x)+exsinx+log5(x)⟶y=xlog(x)+exsinx+log(5)log(x)⟶dxdy=dxd(xlog(x)+exsinx+log(5)log(x))

\begin{gathered} \rm \longrightarrow \dfrac{dy}{dx} = \dfrac{d( \dfrac{log \: x}{x}) }{dx} + \dfrac{d( {e}^{x} \: sin \: x)}{dx} + \dfrac{1}{log \: 5} \dfrac{d(log \: x)}{dx} \\ \\ \\ \rm \longrightarrow \dfrac{dy}{dx} = \frac{x \times \frac{1}{x} - log( x) }{ {x}^{2} } + {e}^{x} \: sin \: x + {e}^{x} cos \: x+ ( \dfrac{1}{x} \times\dfrac{1} {log \: 5} ) \\ \\ \\ \rm \longrightarrow \dfrac{dy}{dx} = \frac{1- log( x) }{ {x}^{2} } + {e}^{x}( \: sin \: x + cos \: x)+ \dfrac{1} {x \: log \: 5} \end{gathered}⟶dxdy=dxd(xlogx)+dxd(exsinx)+log51dxd(logx)⟶dxdy=x2x×x1−log(x)+exsinx+excosx+(x1×log51)⟶dxdy=x21−log(x)+ex(sinx+cosx)+xlog51

Answer :

\rm \dfrac{dy}{dx} = \frac{1- log( x) }{ {x}^{2} } + {e}^{x}( \: sin \: x + cos \: x)+ \dfrac{1} {x \: log \: 5}dxdy=x21−log(x)+ex(sinx+cosx)+xlog51

_________________

d(e^x)/dx = e^x

d(x^n)/dx = n x^(n-1)

d(ln x)/dx = 1/x

d(sin x)/dx = cos x

d(cos x)/dx = - sin x

d(tan x)/dx = sec² x

d(sec x)/dx = tan x * sec x

d(cot x)/dx = - cosec²x

d(cosec x)/dx = - cosec x * cot x