what is the derivative of ( ln ⁡ ( x ) ) x

Francisco Proctor

Francisco Proctor

Answered

2022-07-21

what is the derivative of ( ln ( x ) ) x

Answer & Explanation

Reinfarktq6

Reinfarktq6

Expert

2022-07-22Added 18 answers

Use logarithmic differentiation.
y = ( ln ( x ) ) x
Take ln of both sides: ln ( y ) = x ln ( ln ( x ) )
Now use implicit differentiation: 1 y y = x × 1 ln ( x ) × 1 x + ln ( ln ( x ) ) × 1
y y = 1 ln ( x ) + ln ( ln ( x ) )
Multiply both sides by y:
y = y × [ 1 ln ( x ) + ln ( ln ( x ) ) ]
but we know what y is, thus:
y = ( ln ( x ) ) x × [ 1 ln ( x ) + ln ( ln ( x ) ) ]

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