Answer :

The derivative of f(x) is f'(x) = 42x⁵ * cos⁻¹(x) - 7x⁶ / (√(1 - x²))

To find the derivative of the function f(x) = 7x⁶ * cos⁻¹(x), we'll use the chain rule and the power rule of differentiation.

The chain rule states that if we have a function of the form f(g(x)), then its derivative is given by f'(g(x)) * g'(x).

Let's break down the function f(x) into two parts:

u(x) = 7x⁶

v(x) = cos⁻¹(x)

Now, we'll find the derivatives of u(x) and v(x):

u'(x) = d/dx (7x⁶) = 6 * 7x⁵ = 42x⁵

To find v'(x), we differentiate cos⁻¹(x) using the inverse trigonometric differentiation formula:

d/dx [cos⁻¹(x)] = -1 / (√(1 - x²))

Now, applying the chain rule to the function f(x) = 7x⁶ * cos⁻¹(x), we get:

f'(x) = u'(x) * v(x) + u(x) * v'(x)

f'(x) = 42x⁵ * cos⁻¹(x) + 7x⁶ * (-1 / (√(1 - x²)))

So, the derivative of f(x) is:

f'(x) = 42x⁵ * cos⁻¹(x) - 7x⁶ / (√(1 - x²))

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