Q110 / 160 · TS EAMCET 2021 (06 Aug Shift 1) · Mathematics

If $y=x+\tan x$, then $\cos ^2 x \frac{d^2 y}{d x^2}+2 x$ is equal to

  • A
    -2 y
  • B
    $\frac{2}{3} y$
  • C
    3y
  • Correct answer
    D
    2y

Solution

Given, $y=x+\tan x$ Differentiating, $\frac{d y}{d x}=1+\sec ^2 x$ Again differentiating, $\frac{d^2 y}{d x^2}=0+2 \sec x \sec x \tan x$ $$ =2 \sec ^2 x \cdot \tan x $$ Now, $\cos ^2 x \cdot \frac{d^2 y}{d x^2}+2 x=\cos ^2 x \cdot 2 \sec ^2 x \cdot \tan x+2 x$ $$ =2(x+\tan x)=2 y $$

Hence, the correct answer is option: 'D'.

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