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

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}}) [ \begin{align*} y'&=-\frac{-28x^{3}}{|-7x^{4}|\sqrt{(-7x^{4})^{2}-1}}\ &=\frac{28x^{3}}{|7x^{4}|\sqrt{49x^{8}-1}} \end{align*} ] Since (|7x^{4}| = 7|x^{4}|=7x^{4}) for (x\neq0) [ 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}})