y-log-2-log-3-log-5-x-y-

Question Number 63888 by Mikael last updated on 10/Jul/19

y = {l o g}_{2} [{l o g}_{3} ({l o g}_{5} x)]

y = ?

Answered by Hope last updated on 10/Jul/19

y = {l o g}_{2} [{l o g}_{3} (u)] \to u = {l o g}_{5} x

x = 5^{u} \to \frac{d x}{d u} = 5^{u} l n 5 = x l n 5

y = {l o g}_{2} [v] \to v = {l o g}_{3} u

3^{v} = u

\frac{d u}{d v} = 3^{v} l n 3 = u l n 3 = ({l o g}_{5} x) l n 3

y = {l o g}_{2} v

v = 2^{y}

\frac{d v}{d y} = 2^{y} l n 2 = v l n 2 = 2^{y} l n 2

n o w \frac{d v}{d y} \times \frac{d u}{d v} \times \frac{d x}{d u} = (2^{y} l n 2) \times ({l o g}_{5} x l n 3) \times (x l n 5)

\frac{d x}{d y} = (l n 2 \times l n 3 \times l n 5) ({x l o g}_{5} x) \times (2^{y})

\frac{d x}{d y} = (l n 2 \times l n 3 \times l n 5) \times ({x l n}_{5} x) \times 2^{{l o g}_{2} ({l o g}_{3} ({l o g}_{5} x))}

\frac{d y}{d x} = \frac{1}{(l n 2 \times l n 3 \times l n 5)} \times \frac{1}{{x l n}_{5} x} \times \frac{1}{2^{{l o g}_{2} ({l o g}_{3} ({l o g}_{5} x))}}

p l s c h e c k

Commented by Mikael last updated on 10/Jul/19

t h a n k y o u S i r

Answered by mr W last updated on 10/Jul/19

y^{'} = \frac{1}{\ln 2 \times \log_{3} (\log_{5} x)} \times \frac{1}{\ln 3 \times \log_{5} x} \times \frac{1}{\ln 5 \times x}

y^{'} = \frac{1}{\ln 2 \times \ln 3 \times \ln 5 \times [\log_{3} (\log_{5} x)] [\log_{5} x] x}

Commented by Mikael last updated on 10/Jul/19

T h a n k y o u S i r .

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