Resolved: "For the following exercises, use logarithmic differentiation to..."

William Curry

Answered question

2021-12-25

For the following exercises, use logarithmic differentiation to find

\frac{d y}{d x}

y = {(x^{2} - 1)}^{\ln x}

Donald Cheek

Beginner2021-12-26Added 41 answers

We have given that

y = {(x^{2} - 1)}^{\ln x}

--- 1
Taking logarithm both side of 1 we get

\ln y = {\ln (x^{2} - 1)}^{\ln x}

\Rightarrow \ln y = (\ln x) [\ln (x^{2} - 1)]

Now differentiating both side with respect to x -

\frac{d}{d x} (\ln y) = \frac{d}{d x} [\ln x (\ln (x^{2} - 1))]

\Rightarrow \frac{1}{y} \cdot \frac{d y}{d x} = (\ln x) \frac{d}{d x} [\ln x (x^{2} - 1)] + \ln (x^{2} - 1) \frac{d}{d x} (\ln x)

\Rightarrow \frac{1}{y} \cdot \frac{d y}{d x} = (\ln x) \frac{1}{x^{2} - 1} \cdot \frac{d}{d x} (x^{2} - 1) + \ln (x^{2} - 1) \frac{1}{x}

\Rightarrow \frac{1}{y} \cdot \frac{d y}{d x} = \frac{\ln x}{x^{2} - 1} \cdot (2 x) + \frac{\ln (x^{2} - 1)}{x}

\Rightarrow \frac{d y}{d x} = y [\frac{2 x \ln x}{x^{2} - 1} + \frac{\ln (x^{2} - 1)}{x}]

\Rightarrow \frac{d y}{d x} = {(x^{2} - 1)}^{\ln x} [\frac{2 x \ln x}{x^{2} - 1} + \frac{\ln (x^{2} - 1)}{x}]

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Corgnatiui

Beginner2021-12-27Added 35 answers

Possible derivation:

\frac{d}{d x} ({(- 1 + x^{2})}^{\log (x)})

Express

{(x^{2} - 1)}^{\log (x)}

as a power of

e : {(x^{2} - 1)}^{\log (x)} = e^{\log ({(x^{2} - 1)}^{\log (x)})} = e^{\log (x) \log (x^{2} - 1)} :

= \frac{d}{d x} (e^{\log (x) \log (- 1 + x^{2})})

Using the chain rule,

\frac{d}{d x} (e^{\log (x) \log (x^{2} - 1)}) = \frac{d e^{u}}{d u} \frac{d u}{d x}

, where

u = \log (x) \log (x^{2} - 1)

and

\frac{d}{d u} (e^{u}) = e^{u}

= (\frac{d}{d x} (\log (x) \log (- 1 + x^{2}))) e^{\log (x) \log (- 1 + x^{2})}

Express

e^{\log (x) \log (x^{2} - 1)}

as a power of

x : e^{\log (x) \log (x^{2} - 1)} = e^{\log (x^{\log (x^{2} - 1)})} = x^{\log (x^{2} - 1)} :

= (x^{\log (- 1 + x^{2})}) \frac{d}{d x} (\log (x) \log (- 1 + x^{2}))

Use the product rule,

\frac{d}{d x} (u v) = v \frac{d u}{d x} + u d

user_27qwe Skilled2021-12-30Added 375 answers

First answer is right!

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