Question

como derivar -cosec² de theta elevado ao cubo

Answer 1

$f(\theta)=-\csc^2\left(\theta^2\right)$


Derivando:
$\displaystyle i)~~~~\frac{df}{d\theta}=\frac{df}{du}\frac{du}{dv}\frac{dv}{d\theta}$
(regra da cadeia)
onde:
$\bullet~f=-u^2\\\\\bullet~u=\csc v\\\\\bullet~v=\theta^2$
fazendo a i ficar:
$\displaystyle ii)~~~\frac{df}{d\theta}=-\frac{d}{du}u^2\cdot \frac{d}{dv}\csc v\cdot \frac{d}{d\theta }\theta^2\\\\iii)~~\frac{df}{d\theta}=-2u(-\csc v\cot v)2\theta\\\\iv)~~\frac{df}{d\theta}=4u\theta \csc v\cot v\\\\v)~~\frac{df}{d\theta}=4\theta\csc(\theta^2)\csc(\theta^2)\cot(\theta^2)=\boxed{4\theta\csc^2(\theta^2)\cot(\theta^2)}$

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