Science:Math Exam Resources/Courses/MATH102/December 2015/Question 03/Solution 1

In order to solve this problem, we use implicit differentiation.
Let $y=f^{-1}(x).$
We want to compute ${\frac {dy}{dx}}.$ We can rewrite this equation in the form
$f(y)=x.$
Differentiating both sides, we get
$f'(y){\frac {dy}{dx}}=1$
We then divide to get
${\frac {dy}{dx}}={\frac {1}{f'(y)}}$
We are interested in the point where
$x=1$ and $y=0$.
We plug this in and get
${\frac {dy}{dx}}={\frac {1}{f'(0)}}={\frac {1}{2}}.$
Therefore, we can write the tangent line in point-slope form:
$y={\frac {1}{2}}(x-1)$.
answer: $\color {blue}(c)$