What is the d/(dw)((ln(w))^{(2w)}) ?

The d/(dw)((ln(w))^{(2w)}) is 2ln^{2w}(w)(ln(ln(w))+1/(ln(w)))

What is the first d/(dw)((ln(w))^{(2w)}) ?

The first d/(dw)((ln(w))^{(2w)}) is 2ln^{2w}(w)(ln(ln(w))+1/(ln(w)))

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\frac{d}{d w} ((ln (w))^{(2 w)})

2 ln^{2 w} (w) (ln (ln (w)) + \frac{1}{ln ( w )})

Solution steps

\frac{d}{d w} ((ln (w))^{(2 w)})

Apply exponent rule:

a^{b} = e^{b l n (a)}

(ln (w))^{2 w} = e^{2 w l n (l n (w))}

= \frac{d}{d w} (e^{2 w l n (l n (w))})

Apply the chain rule:

e^{2 w l n (l n (w))} \frac{d}{d w} (2 w ln (ln (w)))

= e^{2 w l n (l n (w))} \frac{d}{d w} (2 w ln (ln (w)))

\frac{d}{d w} (2 w ln (ln (w))) = 2 (ln (ln (w)) + \frac{1}{ln ( w )})

= e^{2 w l n (l n (w))} \cdot 2 (ln (ln (w)) + \frac{1}{ln ( w )})

e^{2 w l n (l n (w))} = ln^{2 w} (w)

= 2 ln^{2 w} (w) (ln (ln (w)) + \frac{1}{ln ( w )})

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What is the d/(dw)((ln(w))^{(2w)}) ?
The d/(dw)((ln(w))^{(2w)}) is 2ln^{2w}(w)(ln(ln(w))+1/(ln(w)))
What is the first d/(dw)((ln(w))^{(2w)}) ?
The first d/(dw)((ln(w))^{(2w)}) is 2ln^{2w}(w)(ln(ln(w))+1/(ln(w)))