To determine

To find: Equation of tangent to the curve.

Answer

Answer: y=23 x- 3 π3+1

Explanation

1) Formula:

i. Constant multiplication rule:

ddxk.fx=k.ddxf(x)

ii. Chain rule: Let Fx=fgx, if g is differentiable at x and f is differentiable at g(x) then F'x=f'gxg'(x)

iii.

ddxsin⁡x=cos⁡x

iv. Equation of tangent line to the curve y=f(x) at point (a,f(a)) is, (y-f(a))=f’(a)(x-a)

2) Given:

y=4 sin2⁡x,      (π/6 ,1)

3) Calculation:

Differentiate y with respect to x,

f'(x)=ddx4 sin2⁡x

By using constant multiplication and chain rule,

ddx4 sin2⁡x=4.ddxsin2⁡x=4.2sin⁡x.ddxsin⁡x=8sin⁡x.cos⁡x

f'x=8sin⁡x.cos⁡x

At (π/6 ,1)

f'π6=8sin⁡π6.cos⁡π6=8.12.32=23 Thus the slope of tangent line at (π/6 ,1) is f'π6=23

Then equation of tangent line to the curve y=4 sin2⁡x at (π/6 ,1) is

(y-1)=23x-π6

y=23 x- 23 π6+1

y=23 x- 3 π3+1

Conclusion:

Equation of tangent to the curve at (π/6 ,1) is y=23 x- 3 π3+1