MCQ
If $y = tan x \,\,tan 2x\,\, tan 3x$ then $\frac{{dy}}{{dx}}$ has the value equal to :
  • A
    $3 sec^2 3x tan x \,\, tan 2x + sec^2 x tan 2x\,\, tan 3x + 2 sec^2 2x\,\,  tan 3x \,\,tan x$
  • B
    $2y (cosec 2x + 2 cosec 4x + 3 cosec 6x)$
  • C
    $3 sec^2 3x - 2 sec^2 2x -sec^2 x$
  • All of the above

Answer

Correct option: D.
All of the above
d
$\operatorname{Tan} 3 x=\tan (2 x+x)$

$\tan 3 x=\frac{\tan 2 x+\tan x}{1-\tan x \tan 2 x}$

$\tan 3 x-\tan x \tan 2 x \tan 3 x=\tan x+\tan 2x$

$y=\operatorname{tan} 3 x-\operatorname{tan} x-\operatorname{tan} 2 x$

$\frac{d y}{d x}=3 \sec ^{2} 3 x-\sec ^{2} x-2 \operatorname{sec}^{2} 2 x$

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Similar questions

Consider the functions defined implicitly by the equation $y^3-3 y+x=0$ on various intervals in the real line. If $x \in(-\infty,-2) \cup(2, \infty)$, the equation implicitly defines a unique real valued differentiable function $y=f(x)$. If $x \in(-2,2)$, the equation implicitly defines a unique real valued differentiable function $y=g(x)$ satisfying $g(0)=0$.

$1.$  If $\mathrm{f}(-10 \sqrt{2})=2 \sqrt{2}$, then $\mathrm{f}^{\prime \prime}(-10 \sqrt{2})=$

$(A)$ $\frac{4 \sqrt{2}}{7^3 3^2}$ $(B)$ $-\frac{4 \sqrt{2}}{7^3 3^2}$ $(C)$ $\frac{4 \sqrt{2}}{7^3 3}$ $(D)$ $-\frac{4 \sqrt{2}}{7^3 3}$

$2.$ The area of the region bounded by the curves $y=f(x)$, the $x$-axis, and the lines $x=a$ and $x=b$, where $-\infty < \mathrm{a} < \mathrm{b} < -2$, is

$(A)$ $\int_a^b \frac{x}{3\left((f(x))^2-1\right)} d x+b f(b)-a f(a)$

$(B)$ $-\int_a^b \frac{x}{3\left((f(x))^2-1\right)} d x+b f(b)-a f(a)$

$(C)$ $\int_a^b \frac{x}{3\left((f(x))^2-1\right)} d x-b f(b)+a f(a)$

$(D)$ $-\int_a^b \frac{x}{3\left((f(x))^2-1\right)} d x-b f(b)+a f(a)$

$3.$ $\int_{-1}^1 g^{\prime}(x) d x=$

$(A)$ $2 g(-1)$ $(B)$ 0 $(C)$ $-2 g(1)$ $(D)$ $2 \mathrm{~g}(1)$

Give the answer question $1,2$ and $3.$

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