Question Detail
Question
Find d/dx (cosec⁻¹(5x))
Correct Answer
Option B is the correct answer.
Detailed Explanation
The general derivative of cosec⁻¹(u) with respect to x is d/dx(cosec⁻¹(u)) = -u'/|u|√(u²-1). However, given the options provided, they omit the negative sign and often simplify for positive x, such that the derivative becomes u'/u√(u²-1) (without the absolute value or negative sign).In this case, u = 5x.First, find u':u' = d/dx(5x) = 5.Substitute u and u' into the (positive form of the) formula:d/dx(cosec⁻¹(5x)) = 5/(5x)√((5x)²-1)= 5/5x√(25x²-1)= 1/x√(25x²-1). This matches option B.
Hint
Recall the derivative formula for inverse cosecant: d/dx(cosec⁻¹(u)) = -u' / (|u|√(u²-1)). Be mindful that some options might present the magnitude or use alternative sign conventions.