Find the derivative of the following function: y ( x ) = ( ln ⁡ ( 2 x ) ) 5 x , x &gt; 1 2

Question

Find the derivative of the following function:       y    (    x    )    =    (    ln    ⁡          (      2      x      )              )              5        x              ,        x    &amp;gt;          1      2

cdtortosadn · Accepted Answer

Whenever you have a function of the form

y = f (x)^{g (x)}

, it is usually a good idea to use logarithmic differentiation:

\log y = g (x) \log f (x),

hence

\frac{d}{d x} [\log y] = \frac{1}{y} \frac{d y}{d x} = \frac{d}{d x} [g (x) \log f (x)],

and

\frac{d y}{d x} = f (x)^{g (x)} (\frac{g (x)}{f (x)} \frac{d f}{d x} + \frac{d g}{d x} \log f (x)) .

So, in your case,

f (x) = \log 2 x

and

g (x) = 5 x

, so we have

\frac{d f}{d x} = \frac{1}{2 x} \cdot 2 = \frac{1}{x},

and

\frac{d g}{d x} = 5,

therefore after minor simplifications,

\frac{d y}{d x} = (\log 2 x)^{5 x} (\frac{5}{\log 2 x} + 5 \log (\log 2 x)) .

The only additional step that was done was to put the fractions over a common denominator of

\log 2 x

and factoring:

\frac{d y}{d x} = 5 (\log 2 x)^{5 x - 1} (1 + (\log 2 x) \log (\log 2 x)) .

Find the derivative of the following function: y(x)=(ln(2x))^(5x),x>1/2

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