Verified answer

1. f'(t)=2cos2t*(-sin2t)*2=-2sin4x   f''=-2cos4x*4=-8cos4x   f''(π/8)=-8*cosπ/2=0

2. f'(x)=4cos³x*(-sinx)+4sin³x*cosx=-4sinx*cosx(cos²x-sin²x)=-2sin2x*cos2x=

-sin4x    f'(π/4)=-sinπ=0

3. f'(x)=1*3/((3x+2)*ln4)   f'(1)=3/5ln4

4. f'(x)=(1/2)*(5x-10x²)^(-1/2) *(5-20x)=(5-20x)/(2√(5x-10x²))

5. f'(x)=e^(cos2x) *(-sin2x)*2=-2sin2x*e^(cos2x)   f'(π/4)=-2sin(π/2)*e^(cos(π/2)=

-2

6. y=2x²-12x+16     x₀=5

y(5)=50-60+16=6

y'=4x-12    y'(5)=20-12=8

y=y(5)+y'(5)(x-5)=6+8(x-5)=6+8x-40=8x-34     y=8x-34