Answer:
9. y = x - 1/2 + π/4
10. y = (-2/3)(√6)x + (2/3)√3 + π/3
Explanation:
9. From a table of derivatives,
... arctan'(x) = 1/(1+x²)
Then, using the chain rule, ...
... f'(x) = arctan'(2x) = 2/(1 +(2x)²) = 2/(1 +4x²)
For x=1/2, the slope of the tangent is ...
... f'(1/2) = 2/(1 +4·(1/2)²) = 1
Then in point-slope form, the equation of the tangent line is
... y = 1(x -1/2) +π/4
... y = x -1/2 +π/4
10. From a table of derivatives,
... arccos'(x) = -1/√(1 -x²)
Then, using the chain rule, ...
... arccos'(x²) = -2x/√(1 -x⁴)
For x=1/√2, the slope of the tangent is ...
... f'(1/√2) = -(2/√2)/√(1 -(1/√2)⁴) = (-√2)/√(3/4) = -2√(2/3)
... f'(1/√2) = (-2/3)√6
Then in point-slope form, the equation of the tangent line is
... y = (-2/3)(√6)(x -1/√2) +π/3
... y = (-2/3)(√6)x +(2/3)√3 +π/3