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For all questions below use the Chain Rule to find the derivative
dx/d(f(g(x)) = f'(g(x)) • g'(x)
1.Find the derivative of the function y = tan (1/x)
2.Suppose that f' (4) = g(4) = g' (4) = 1. Do we
have enough information to compute F' (4), where F(x) = f(g(x))? If not, what is missing?
3. Find the equation of the tangent line of the function y = √ 4 - 3 cos(x) at the point (0, 1)

Answer 1

To find the derivative of the function
`y = tan (1/x)`the chain rule. We differentiate the outer function, tan(x), and the inner function,
`1/x`separately, and then multiply the results.

Step-by-step explanation:

To find the derivative of the function y = tan (1/x), we can use the chain rule. Let's start by defining
`f(x) = tan(x)` The Chain Rule states that the derivative of f(g(x)) is equal to
`f'(g(x)) * g'(x)`

To find f'(g(x)), we differentiate f(x) with respect to x and evaluate it at g(x). The derivative of tan(x) is
`sec^2(x), so f'(g(x))`

To find g'(x), we differentiate g(x) with respect to x. Using the power rule, we get
`g'(x) = -1/x^2.`

Now, we can apply the chain rule:
`dx/d(f(g(x))) = f'(g(x)) * g'(x) = sec^2(g(x)) * (-1/x^2).`