Question Detail
Question
Find the first derivative of sec⁻¹(5x) with respect to x.
Correct Answer
Option D is the correct answer.
Detailed Explanation
The derivative of sec⁻¹(u) with respect to x is d/dx(sec⁻¹(u)) = (1 / (|u|√(u² - 1))) * du/dx.In this problem, u = 5x.So, du/dx = d/dx(5x) = 5.Substitute u and du/dx into the formula:d/dx(sec⁻¹(5x)) = (1 / (|5x|√((5x)² - 1))) * 5= 5 / (5|x|√(25x² - 1))= 1 / (|x|√(25x² - 1)).For the purpose of these options, it's typically assumed x > 0, so |x| = x.= 1 / (x√(25x² - 1)).
Hint
Recall the differentiation formula for inverse secant: d/dx(sec⁻¹(u)) = (1 / (|u|√(u² - 1))) * du/dx.