Question

Suppose f(x)=4x^5 - (1/x^4). Find the slope of the tangent line to the graph of y = f^(–1)(x) at (3, 1)

Answer 1

1/24
1/16
00062
1620
972

Answer 2

@iambatman

Answer 3

@Zale101

Answer 4

there is a theorem for finding derivative of inverse

Answer 5

$$
\Large (f^{-1}(x))' = \frac{1}{f' (f^{-1}(x))}

$$

Answer 6

do you see how to use it ?

Answer 7

When I took the inverse of the function and then derived it I got: (20x^9 + 4)/ x^5

Answer 8

is this the original function
$$
\Large f(x) = 4x^5 - \frac{1}{x^4}
$$

Answer 9

Yes.

Answer 10

Yes.

Answer 11

and can you show me how you got the inverse, did you switch x and y ?

Answer 12

Oh I think I derived the original function and not the inverse.

Answer 13

it is actually not necessary to find the inverse function explicitly , because of this theorem
$$
\Large{
(f^{-1}(x))' = \frac{1}{f' (f^{-1}(x))}
\\ But...\\
f(x) =4x^5 - \frac{1}{x^4} = 4x^5 - x^{-4}
\\
f ' (x) = 4(5)x^4 - (-4)x^{-5} = 20x^4 + \frac{4}{x^5}

\\ \therefore \\
(f^{-1}(3))' = \frac{1}{f' (f^{-1}(3))} = \frac{1}{f' (1)} = \frac{1}{20\cdot 1 + \frac{4}{1}}

}
$$

Answer 14

Wow, using the formula makes it so much easier. Thank you so much for the help!