Key to "Derivatives of logarithmic functions Calculate the derivative of..."

banganX

Answered question

2021-10-14

Derivatives of logarithmic functions Calculate the derivative of the following functions

y = \frac{1}{\log_{4} x}

Answer & Explanation

opsadnojD

Skilled2021-10-15Added 95 answers

Step 1
To find:
The derivative of $y = \frac{1}{\log_{4} x}$ .
Concept used:
Chain rule of derivative:
$(f (g (x)))^{'} = f^{'} (g (x)) * g^{'} (x)$
Power rule of derivative:
${(x^{n})}^{'} = n \cdot x^{n - 1}$
The derivative of ${(\log_{a} x)}^{'} = \frac{1}{x \ln a}$ .
power rule of logarithm:
${\log a}^{b} = b \cdot \ln a$
Step 2
Calculation:
Differentiate $y = \frac{1}{\log_{4} x}$ with respect to x.
$\frac{d y}{d x} = \frac{d}{d x} ({(\log_{4} x)}^{- 1}) \cdot \frac{d}{d x} (\log_{4} x)$
$= - {(\log_{4} x)}^{- 1 - 1} \cdot \frac{1}{x \ln 4}$
$= - {(\log_{4} x)}^{- 2} \cdot \frac{1}{x {\ln 2}^{2}}$
$= - \frac{1}{2 x \cdot {(\log_{4} x)}^{2} \cdot \ln 2}$
Therefore, the derivative of $y = \frac{1}{\log_{4} x} i s \frac{d y}{d x} = - \frac{1}{2 x \cdot {(\log_{4} x)}^{2} \cdot \ln 2}$ .

Jeffrey Jordon

Expert2022-08-15Added 2605 answers

Answer is given below (on video)

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