EQUATION OF TANGENT LINE TO INVERSE FUNCTION

Consider the function f(x). Let g(x) be the inverse function of f(x).

g(x) = f^-1(x)

To find the slope of a tangent line to g(x), we have to find the first derivative of g(x), that is g'(x). Then, find the value of g'(x) at the point of tangency.

Since g(x) is the inverse of f(x), we can find the derivative of g(x) as follows. 

g(x) = f^-1(x)

Take the function f on both sides using the operation function composition.

f ∘ g(x)] = f ∘ f^-1(x)

f[g(x)] = f[f^-1(x)]

f[g(x)] = f^1-1(x)

f[g(x)] = f⁰(x)

f[g(x)] = x

Find the derivative on both sides with respect to x. (Use chain rule on the left side. That is, first find the derivative of f, then by chain rule, find the derivative of g(x)).

f'[g(x)] ⋅ g'(x) = 1

Divide both sides by f'[g(x)].

Let g(5) = k.

Since g(x) is the inverse of f(x),

f(k) = 5

k³ + k + 5 = 5

Subtract 5 from both sides.

k³ + k = 0

k(k² + 1) = 0

Therefore,

k = 0

Since g(5) =k,

g(5) = 0

So, the point on g(x) corresponding to x = 5 is (5, 0).

f(x) = x³ + x + 5

f'(x) = 3x² + 1

Substitute x = 0.

f'(0) = 3(0)² + 1

f'(0) = 0 + 1

f'(0) = 1

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