Inverse Function Tangents

00:00 5 marks

Let f be a differentiable function with a non-zero derivative and let f^{-1} denote its inverse. Prove that the derivative of f^{-1} at the point y = f(a) is given by \( (f^{-1})'(f(a)) = \frac{1}{f'(a)} \).

Then, if \( f(x) = 2x + 5 \), find the slope of the tangent line to \( f^{-1} \) at \( x = 9 \).

Hint Hint

Hint

Recall that differentiating the identity \( f(f^{-1}(x)) = x \) provides the reciprocal relationship between the derivatives.

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation
Common Errors Common Errors
Real World Context Real World Context
Focus Focus
Formula Booklet

Answer Answer

Complete Answer

For \( f(x) = 2x + 5 \), we have \( (f^{-1})'(9) = \frac{1}{2} \).

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation

Common Errors Common Errors

Focus Focus

Formula Booklet