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