Задача 10 (Похідна степенево-показникової функції)

Обчислити похідну степенево-показникової функції  y = \left(x^{3}+4 \right)^{tgx}

 y=U^{V},\; U=x^{3}+4,\; V=tgx

 \left(U^{V} \right)'=U^{V}\left(V'lnU+V\cdot \left(lnU \right)' \right)

 y'=\left(x^{3}+4 \right)^{tgx}\left(\left(tgx \right)'ln\left(x^{3}+4 \right)+tgx\left(ln(x^{3}+4) \right)'\right)=

 =\left(x^{3}+4 \right)^{tgx}\left(\frac{1}{cos^{2}x} \cdot ln\left(x^{3}+4 \right)+tgx\cdot \frac{1}{x^{3}+4}\cdot \left(x^{3}+4 \right)'\right)=

 =\left(x^{3}+4 \right)^{tgx}\left(\frac{ln\left(x^{3}+4 \right)}{cos^{2}x}+\frac{3x^{2}tgx}{x^{3}+4} \right)

One Response to “Задача 10 (Похідна степенево-показникової функції)”

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