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Derivative of an inverse function practice problems

(f⁻¹)′(a) = 1 / f′(f⁻¹(a)), without ever solving for the inverse. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.

Let \( \displaystyle f(x) = 2 x + \sin{\left(x \right)} \), which is one-to-one. Find \( \displaystyle (f^{-1})'(0) \).

Problem 2.1545medium✓ Nihil obstat

Let \( \displaystyle f(x) = x^{5} + x \), which is one-to-one. Find \( \displaystyle (f^{-1})'(0) \).

Problem 2.1546medium✓ Nihil obstat

Let \( \displaystyle f(x) = x + e^{x} \), which is one-to-one. Find \( \displaystyle (f^{-1})'(1) \).

Problem 2.1547medium✓ Nihil obstat

Let \( \displaystyle f(x) = x^{3} + x^{2} + 2 x \), which is one-to-one. Find \( \displaystyle (f^{-1})'(-8) \).

Problem 2.1548medium✓ Nihil obstat

Let \( \displaystyle f(x) = x^{3} + 4 x + 2 \), which is one-to-one. Find \( \displaystyle (f^{-1})'(-14) \).

Problem 2.1549medium✓ Nihil obstat

Let \( \displaystyle f(x) = 2 x + e^{x} \), which is one-to-one. Find \( \displaystyle (f^{-1})'(1) \).

Problem 2.1550medium✓ Nihil obstat

Let \( \displaystyle f(x) = x^{5} + 3 x \), which is one-to-one. Find \( \displaystyle (f^{-1})'(38) \).

Problem 2.1551medium✓ Nihil obstat

Let \( \displaystyle f(x) = x^{3} + x^{2} + 4 x \), which is one-to-one. Find \( \displaystyle (f^{-1})'(-4) \).

Problem 2.1552medium✓ Nihil obstat

Let \( \displaystyle f(x) = x^{3} + x^{2} + 2 x \), which is one-to-one. Find \( \displaystyle (f^{-1})'(-2) \).

Problem 2.1553medium✓ Nihil obstat

Let \( \displaystyle f(x) = x^{3} + x^{2} + 3 x \), which is one-to-one. Find \( \displaystyle (f^{-1})'(-3) \).

Problem 2.1554medium✓ Nihil obstat