Find the Derivative - d/dx arcsec(x)

Problem

d(arcsec(x))/d(x)

Solution

1. Set up the equation by letting y=arcsec(x) which implies x=sec(y) for y in the range [0,π/2)∪(π/2,π]

2. Differentiate implicitly both sides of x=sec(y) with respect to x

d(x)/d(x)=d(sec(y))/d(x)

3. Apply the chain rule to the right side of the equation.

1=sec(y)*tan(y)d(y)/d(x)

4. Solve for the derivative d(y)/d(x) by isolating it.

d(y)/d(x)=1/(sec(y)*tan(y))

5. Use the trigonometric identity tan2(y)+1=sec2(y) to express tan(y) in terms of sec(y)

tan(y)=±√(,sec2(y)−1)

6. Substitute x back into the expression using x=sec(y)

d(y)/d(x)=1/(x√(,x2−1))

7. Account for the sign by noting that the slope of arcsec(x) is always positive where defined, requiring the use of the absolute value for x

d(y)/d(x)=1/(|x|√(,x2−1))

Final Answer

d(arcsec(x))/d(x)=1/(|x|√(,x2−1))

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