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in Continuity and Differentiability by (28.2k points)
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(i) Match the following.

A B
\(\frac{d}{dx}(x)\) 0
\(\frac{d}{dx}(8)\) \(\frac{1}{2\sqrt{x}}\)
\(\frac{d}{dx}cos^{-1}x\) 1
\(\frac{d}{dx}\sqrt{x}\) \(\frac{-1}{\sqrt{1-x^2}}\)
2\(\sqrt{x}\)

(ii) If y = ea cos-1x, then show that (1 – x2)y2 – xy1 – a2y = 0 

1 Answer

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Best answer

(i) 

 A B
\(\frac{d}{dx}(x)\) 1
\(\frac{d}{dx}(8)\)  0
\(\frac{d}{dx}cos^{-1}x\) \(\frac{-1}{\sqrt{1-x^2}}\)
\(\frac{d}{dx}\sqrt{x}\) \(\frac{1}{2\sqrt{x}}\)

 (ii) Given, 

y = ea cos-1x ____(1)

Differentiating w.r.to x,

Again differentiating w. r. t to x

⇒ (1 – x2)y2 – xy1 = a2. ea cos-1x

⇒ (1 – x2)y2 – xy1 = a2y

⇒ (1 – x2)y2 – xy1 – a2y = 0.

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