Question

determine \\(\frac{dy}{dx}\\) for \\(y = \cos^{-1}(8x^3)\\).

Explanation:

Apply the chain rule for inverse cosine

Using the Inverse Trigonometric Derivatives and Chain Rule Differentiation knowledge points

$$ \frac{d}{dx}[\cos^{-1}(u)] = -\frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx} $$

$$ u = 8x^3 $$

Differentiate the inner function

Using the Power Rule Differentiation knowledge point

$$ \frac{du}{dx} = \frac{d}{dx}[8x^3] = 24x^2 $$

Substitute and simplify the derivative

Using the Inverse Trigonometric Derivatives and Chain Rule Differentiation knowledge points

$$ \frac{dy}{dx} = -\frac{1}{\sqrt{1-(8x^3)^2}} \cdot 24x^2 = -\frac{24x^2}{\sqrt{1-64x^6}} $$

Answer:

Determine \(\frac{dy}{dx}\) for \(y = \cos^{-1}(8x^3)\).

\(\frac{dy}{dx} =\) <blank>\(-\frac{24x^2}{\sqrt{1-64x^6}}\)</blank>