If y=Sin x ⋅ Sin 2x ⋅ Sin 3 x ldots Sin n x, then y' is

Question

If y=Sin x ⋅ Sin 2x ⋅ Sin 3 x ldots Sin n x, then y' is (A)  displaystyle∑k=1n k tan k x (B) y . displaystyle∑k=1n k cot kx (C) y. displaystyle∑k=1n k tan kn (D)  displaystyle∑k=1n cot kx

Tardigrade Admin · Accepted Answer

Given, y= sin x ⋅ sin 2 x ⋅ sin 3 x ⋅ ldots sin n x Taking log on both sides, log y= log sin x+ log sin 2 x+ ldots+ log sin n x Differentiating w.r.t. x (1/y) ⋅ (d y/d x)=1 ⋅ cot x+2 cot 2 x+ ldots +n cot n x ⇒ (d y/d x)= y ⋅ displaystyle ∑k=1n k cot k x ⇒ y'=y ⋅ displaystyle ∑k=1n k cot k x

Q. If $y = S in x \cdot S in 2 x \cdot S in 3 x \dots S in n x$ , then y' is

$k = 1 \sum n k t an k x$

$y . k = 1 \sum n k co t k x$

$y . k = 1 \sum n k t an kn$

$k = 1 \sum n co t k x$

Q. If y=Sinx⋅Sin2x⋅Sin3x…Sinnx, then y' is

k=1∑n​ktankx

y.k=1∑n​kcotkx

y.k=1∑n​ktankn

k=1∑n​cotkx

Given, y=sinx⋅sin2x⋅sin3x⋅…sinnx Taking log on both sides, logy=logsinx+logsin2x+…+logsinnx Differentiating w.r.t. x y1​⋅dxdy​=1⋅cotx+2cot2x+…+ncotnx ⇒dxdy​=y⋅k=1∑n​kcotkx ⇒y′=y⋅k=1∑n​kcotkx

Q. If $y = S in x \cdot S in 2 x \cdot S in 3 x \dots S in n x$ , then y' is

$k = 1 \sum n k t an k x$

$y . k = 1 \sum n k co t k x$

$y . k = 1 \sum n k t an kn$

$k = 1 \sum n co t k x$