"Evaluate the following derivatives. ddx(x−ln⁡x)" was answered by students like you

Khaleesi Herbert

Answered question

2021-02-16

Evaluate the following derivatives.

\frac{d}{d x} (x^{- \ln x})

Answer & Explanation

StrycharzT

Skilled2021-02-17Added 102 answers

Step 1
To evaluate the derivatives:

\frac{d}{d x} (x^{- \ln x})

Solution:
Let

y = x^{- \ln x}

Taking logarithm on both sides,

\ln y = {\ln x}^{- \ln x}

\ln y = - \ln x \cdot \ln x

\ln y = - {(\ln x)}^{2}

Differentiating with respect to x,

\frac{1}{y} \cdot \frac{d y}{d x} = - 2 \ln x \cdot \frac{1}{x}

\frac{d y}{d x} = - 2 y \frac{\ln x}{x}

\frac{d y}{d x} = \frac{- 2 x^{- \ln x} \ln x}{x}

Step 2
Hence, required derivative is

\frac{d y}{d x} = \frac{- 2 x^{- \ln x} \ln x}{x}

.

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