3.13-bo'lim : Logarifmik farqlash
4. \(g\left( w \right) = {\left( {3w - 7} \right)^{4w}}\) ning birinchi hosilasini toping.
Yechim
Biz faqat bir oz logarifmik farqlashimiz kerak, shuning uchun ikkala tomonning logarifmini oling va biroz soddalashtiring.
\[\ln \left[ {g\left( w \right)} \right] = \ln \left[ {{{\left( {3w - 7} \right)}^{4w}}} \right] = 4w\,\,\ln \left( {3w - 7} \right)\]2-qadam
\(w\) ga nisbatan har ikki tomonni farqlash uchun yashirin farqlashdan foydalaning. Mahsulotni o'ng tomonda boshqarishni unutmang.
\[\frac{{g'\left( w \right)}}{{g\left( w \right)}} = 4\ln \left( {3w - 7} \right) + 4w\frac{3}{{3w - 7}} = 4\ln \left( {3w - 7} \right) + \frac{{12w}}{{3w - 7}}\]3-qadam
Nihoyat, lotinni yeching va \(g\left( w \right)\) uchun tenglamani kiriting.
\[\begin{align*}g'\left( w \right) & = g\left( w \right)\left[ {4\ln \left( {3w - 7} \right) + \frac{{12w}}{{3w - 7}}} \right]\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{{{\left( {3w - 7} \right)}^{4w}}\left[ {4\ln \left( {3w - 7} \right) + \frac{{12w}}{{3w - 7}}} \right]}}\end{align*}\]