Question

determine \\(\frac{dy}{dx}\\) for \\(y = 6x^7 \sin^{-1} x\\).

Explanation:

Identify the function and differentiation rules

The given function is:

$$ y = 6x^7 \sin^{-1}(x) $$

This is a product of two functions: \(u(x) = 6x^7\) and \(v(x) = \sin^{-1}(x)\).

Differentiate the individual components

Using the power rule for \(u(x)\):

$$ u'(x) = 42x^6 $$

Using the derivative of the arcsine function for \(v(x)\):

$$ v'(x) = \frac{1}{\sqrt{1-x^2}} $$

Apply the product rule

The product rule states:

$$ \frac{dy}{dx} = u'(x)v(x) + u(x)v'(x) $$

Substituting the derivatives:

$$ \frac{dy}{dx} = 42x^6 \sin^{-1}(x) + \frac{6x^7}{\sqrt{1-x^2}} $$

Answer:

Determine \(\frac{dy}{dx}\) for \(y = 6x^7\sin^{-1}x\).

\(\frac{dy}{dx} =\) <blank>\(42x^6 \sin^{-1}(x) + \frac{6x^7}{\sqrt{1-x^2}}\)</blank>