Question

Differentiate (x+1/x)^x with respect to x.​

Answer 1

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Use logarithmic differentiation to get

y'=(lnx+1−1x)x

x−1

.

Explanation:

Take the natural logarithm of both sides, to drop the exponent:

ln

y

=

(

x

−

1

)

ln

x

Now differentiate both sides with respect to

x

:

d

d

x

(

ln

y

=

(

x

−

1

)

ln

x

)

Note that this will require knowledge of implicit differentiation, because the derivative of

ln

y

w.r.t.x is

1

y

⋅

y

'

. The derivative of

(

x

−

1

)

ln

x

is found with the product rule:

d

d

x

(

(

x

−

1

)

(

ln

x

)

)

=

(

x

−

1

)

'

(

ln

x

)

+

(

x

−

1

)

(

ln

x

)

'

=

(

1

)

(

ln

x

)

+

(

x

−

1

)

⋅

(

1

x

)

=

ln

x

+

x

−

1

x

=

ln

x

+

1

−

1

x

Since

d

d

x

(

ln

y

)

=

1

y

⋅

y

'

, and

d

d

x

(

(

x

−

1

)

(

ln

x

)

)

=

ln

x

+

1

−

1

x

, we have:

1

y

⋅

y

'

=

ln

x

+

1

−

1

x

Multiply both sides by

y

to isolate

y

'

:

y

'

=

(

ln

x

+

1

−

1

x

)

y

Since

y

=

x

x

−

1

:

y

'

=

(

ln

x

+

1

−

1

x

)

x

x

−

1

Answer 2

Step-by-step explanation:

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