If f(x) = 7x^6cos^-1x, find f '(x).

y = 7 x^6 cos^-1 x

f' = 7[ x^6 (1/(1-x^2)) + 6x^5 cos^-1 x ]

= 7 x^6/(1-x^2) + 42x^5 cos^-1 x

d/dx cos^-1(x) = -1/√(1-x^2)

To find f'(x), the derivative of f(x), we can use the product rule and the chain rule.

First, let's apply the product rule. The product rule states that if we have two functions u(x) and v(x), the derivative of their product is given by:

(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)

In this case, u(x) = 7x^6 and v(x) = cos^(-1)x.

Taking the derivative of u(x), we get:

u'(x) = d/dx (7x^6) = 42x^5

Now, to find the derivative of v(x), we need to use the chain rule. The chain rule states that if we have a composition of functions (f(g(x))), then the derivative is given by:

d/dx (f(g(x)) = f'(g(x)) * g'(x)

In this case, f(x) = cos^(-1)x, so g(x) = x and f'(x) = d/dx (cos^(-1)x).

To find f'(x), let's use the inverse trigonometric derivative formula:

d/dx (cos^(-1)x) = -1 / √(1 - x^2)

Now we have all the components to find f'(x):

f'(x) = u'(x) * v(x) + u(x) * v'(x)
= 42x^5 * cos^(-1)x + 7x^6 * (-1 / √(1 - x^2))
= 42x^5 * cos^(-1)x - 7x^6 / √(1 - x^2)

Therefore, f'(x) = 42x^5 * cos^(-1)x - 7x^6 / √(1 - x^2).

To find the derivative of the function f(x) = 7x^6cos^-1x, we can use the product and chain rules.

1. The product rule states that if we have two functions u(x) and v(x), the derivative of their product uv with respect to x is given by:
(uv)' = u'v + uv'

2. The chain rule states that if we have a composition of functions f(g(x)), the derivative of this composition with respect to x is given by:
(f(g(x)))' = f'(g(x)) * g'(x)

Now let's find the derivative of f(x).

Step 1: Find the derivative of 7x^6.
The derivative of 7x^6 with respect to x is:
(7x^6)' = 6 * 7x^5 = 42x^5

Step 2: Find the derivative of cos^-1x.
The derivative of cos^-1x with respect to x is given by the chain rule:
(cos^-1x)' = -1 / sqrt(1 - x^2)

Step 3: Apply the product rule to find f'(x).
Using the product rule, we have:
f'(x) = (7x^6)' * cos^-1x + 7x^6 * (cos^-1x)'

Substituting the derivatives we obtained in the previous steps, we get:
f'(x) = 42x^5 * cos^-1x + 7x^6 * (-1 / sqrt(1 - x^2))

Therefore, the derivative of f(x) = 7x^6cos^-1x is f'(x) = 42x^5cos^-1x - 7x^6 / sqrt(1 - x^2).