Question

1. $y = csc^{-1}(-7x^{4})$

Explanation:

Step1: Recall the derivative formula for \(y = \csc^{-1}(u)\)

The derivative of \(y=\csc^{-1}(u)\) with respect to \(x\) is \(y'=-\frac{u'}{|u|\sqrt{u^{2}-1}}\), where \(u = - 7x^{4}\)

Step2: Find the derivative of \(u=-7x^{4}\)

Using the power rule \((x^{n})'=nx^{n - 1}\), we have \(u'=-28x^{3}\)

Step3: Substitute \(u\) and \(u'\) into the derivative formula

Substitute \(u=-7x^{4}\) and \(u'=-28x^{3}\) into \(y'=-\frac{u'}{|u|\sqrt{u^{2}-1}}\)

$$
LATEXBLOCK0
$$

Since \(|7x^{4}| = 7|x^{4}|=7x^{4}\) for \(x
eq0\)

$$ y'=\frac{28x^{3}}{7x^{4}\sqrt{49x^{8}-1}}=\frac{4}{x\sqrt{49x^{8}-1}} $$

Answer:

\(y'=\frac{4}{x\sqrt{49x^{8}-1}}\)