find the derivative of y with respect to x.\n\n$y = \\cos^{-1}(8x^{3})$
find the derivative of y with respect to x.\n\n$y = \\cos^{-1}(8x^{3})$
Answer
Explanation:
Step1: Apply the chain rule
The chain rule states that if (y = f(g(x))), then (y^\prime=f^\prime(g(x))\cdot g^\prime(x)). For (y = \cos^{- 1}(u)) where (u = 8x^{3}), the derivative of (\cos^{-1}(u)) with respect to (u) is (-\frac{1}{\sqrt{1 - u^{2}}}), and the derivative of (u = 8x^{3}) with respect to (x) is (24x^{2}).
Step2: Substitute (u = 8x^{3}) into the formula
[ \begin{align*} \frac{dy}{dx}&=-\frac{1}{\sqrt{1-(8x^{3})^{2}}}\cdot(8x^{3})^\prime\ &=-\frac{1}{\sqrt{1 - 64x^{6}}}\cdot24x^{2}\ &=-\frac{24x^{2}}{\sqrt{1 - 64x^{6}}} \end{align*} ]
Answer:
(-\frac{24x^{2}}{\sqrt{1 - 64x^{6}}})