#### 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.

ddxsinx=cosx

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 sin2x, (π/6 ,1)**

**3) Calculation:**

Differentiate y with respect to x,

f'(x)=ddx4 sin2x

By using constant multiplication and chain rule,

ddx4 sin2x=4.ddxsin2x=4.2sinx.ddxsinx=8sinx.cosx

f'x=8sinx.cosx

At (π/6 ,1)

f'π6=8sinπ6.cosπ6=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 sin2x 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