Question

$\bf \:Differnatiate \: {e}^{ log(x + \sqrt{ {x}^{2} + {a}^{2} } ) } w.r.t. \: x$​

Answer 1

$\Large{\underbrace{\sf{\purple{Required\:Solution:}}}}$

$\sf \:Let \: y = {e}^{ log(x + \sqrt{ {x}^{2} + {a}^{2} } ) }$

$\sf\implies \:y = log(x + \sqrt{ {x}^{2} + {a}^{2} } )$

$\sf \:Differnatiate \: w.r.t. \: x$

$\sf\implies \:\dfrac{dy}{dx} = \dfrac{d}{dx} log(x + \sqrt{ {x}^{2} + {a}^{2} } )$

$\sf\implies \:\dfrac{dy}{dx} = \dfrac{1}{x + \sqrt{ {x}^{2} + {a}^{2} } } \dfrac{d}{dx}( x + \sqrt{ {x}^{2} + {a}^{2} } )$

$\sf\implies \:\dfrac{dy}{dx} = \dfrac{1}{x + \sqrt{ {x}^{2} + {a}^{2} } } (1 + \dfrac{1}{2 \sqrt{ {x}^{2} + {a}^{2} } } \dfrac{d}{dx} ( {x}^{2} + {a}^{2} )$

$\sf\implies \:\dfrac{dy}{dx} = \dfrac{1}{x + \sqrt{ {x}^{2} + {a}^{2} } } (1 + \dfrac{1}{2 \sqrt{ {x}^{2} + {a}^{2} } }(2x))$

$\sf\implies \:\dfrac{dy}{dx} = \dfrac{1}{x + \sqrt{ {x}^{2} + {a}^{2} } } (1 + \dfrac{x}{ \sqrt{ {x}^{2} + {a}^{2} } })$

$\sf\implies \:\dfrac{dy}{dx} = \dfrac{1}{x + \sqrt{ {x}^{2} + {a}^{2} } } ( \dfrac{x + \sqrt{ {x}^{2} + {a}^{2} } }{ \sqrt{ {x}^{2} + {a}^{2} } })$

$\sf\implies \:\dfrac{dy}{dx} = \dfrac{1}{ \sqrt{ {x}^{2} + {a}^{2} } }$

___________________________

Formula Used:-

• $\sf \:\dfrac{d}{dx} {x}^{n } = n {x}^{n - 1}$
• $\sf \:\dfrac{d}{dx} log(x) = \dfrac{1}{x}$
• $\sf \:\dfrac{d}{dx} \sqrt{x} = \dfrac{1}{2 \sqrt{x} }$
• $\sf \:\dfrac{d}{dx}k = 0$
• $\sf \: {e}^{ log(x) } = x$

________________________________

Answer 2

Step-by-step explanation:

Lety=e

log(x+

x

2

+a

2

)

\begin{gathered}\sf\implies \:y = log(x + \sqrt{ {x}^{2} + {a}^{2} } ) \\\\\end{gathered}

⟹y=log(x+

x

2

+a

2

)

\begin{gathered}\sf \:Differnatiate \: w.r.t. \: x\\\\\end{gathered}

Differnatiatew.r.t.x

\begin{gathered}\sf\implies \:\dfrac{dy}{dx} = \dfrac{d}{dx} log(x + \sqrt{ {x}^{2} + {a}^{2} } ) \\\\\end{gathered}

⟹

dx

dy

=

dx

d

log(x+

x

2

+a

2

)

\begin{gathered}\sf\implies \:\dfrac{dy}{dx} = \dfrac{1}{x + \sqrt{ {x}^{2} + {a}^{2} } } \dfrac{d}{dx}( x + \sqrt{ {x}^{2} + {a}^{2} } )\\\\\end{gathered}

⟹

dx

dy

=

x+

x

2

+a

2

1

dx

d

(x+

x

2

+a

2

)

\begin{gathered}\sf\implies \:\dfrac{dy}{dx} = \dfrac{1}{x + \sqrt{ {x}^{2} + {a}^{2} } } (1 + \dfrac{1}{2 \sqrt{ {x}^{2} + {a}^{2} } } \dfrac{d}{dx} ( {x}^{2} + {a}^{2} )\\\\\end{gathered}

⟹

dx

dy

=

x+

x

2

+a

2

1

(1+

2

x

2

+a

2

1

dx

d

(x

2

+a

2

)

\begin{gathered}\sf\implies \:\dfrac{dy}{dx} = \dfrac{1}{x + \sqrt{ {x}^{2} + {a}^{2} } } (1 + \dfrac{1}{2 \sqrt{ {x}^{2} + {a}^{2} } }(2x))\\\\\end{gathered}

⟹

dx

dy

=

x+

x

2

+a

2

1

(1+

2

x

2

+a

2

1

(2x))

\begin{gathered}\sf\implies \:\dfrac{dy}{dx} = \dfrac{1}{x + \sqrt{ {x}^{2} + {a}^{2} } } (1 + \dfrac{x}{ \sqrt{ {x}^{2} + {a}^{2} } })\\\\\end{gathered}

⟹

dx

dy

=

x+

x

2

+a

2

1

(1+

x

2

+a

2

x

)

\begin{gathered}\sf\implies \:\dfrac{dy}{dx} = \dfrac{1}{x + \sqrt{ {x}^{2} + {a}^{2} } } ( \dfrac{x + \sqrt{ {x}^{2} + {a}^{2} } }{ \sqrt{ {x}^{2} + {a}^{2} } })\\\\\end{gathered}

⟹

dx

dy

=

x+

x

2

+a

2

1

(

x

2

+a

2

x+

x

2

+a

2

)

\begin{gathered}\sf\implies \:\dfrac{dy}{dx} = \dfrac{1}{ \sqrt{ {x}^{2} + {a}^{2} } } \\\\\end{gathered}

⟹

dx

dy

=

x

2

+a

2

1

___________________________

Formula Used:-

\begin{gathered}\sf \:\dfrac{d}{dx} {x}^{n } = n {x}^{n - 1} \\\\\end{gathered}

dx

d

x

n

=nx

n−1

\begin{gathered}\sf \:\dfrac{d}{dx} log(x) = \dfrac{1}{x}\\\\ \end{gathered}

dx

d

log(x)=

x

1

\begin{gathered}\sf \:\dfrac{d}{dx} \sqrt{x} = \dfrac{1}{2 \sqrt{x} }\\\\ \end{gathered}

dx

d

x

=

2

x

1

\begin{gathered}\sf \:\dfrac{d}{dx}k = 0\\\\\end{gathered}

dx

d

k=0

\begin{gathered}\sf \: {e}^{ log(x) } = x\\\\\end{gathered}

e

log(x)

=x