**Question 1. If ****, prove that **

**Solution:**

We have,

⇒

Squaring both sides, we get,

y

^{2}= x + y

**Question 2. If ****, prove that **

**Solution:**

We have,

⇒

Squaring both sides, we get,

y

^{2}= cos x + y⇒

**Question 3. If ****, prove that **

**Solution:**

We have,

⇒

Squaring both sides, we get,

y

^{2}= log x + y

**Question 4. If **** , prove that **

**Solution:**

We have,

⇒

Squaring both sides, we get,

y

^{2}= tan x + y

**Question 5. If **** , prove that **

**Solution:**

We have,

⇒ y = (sin x)

^{y}Taking log on both sides,

log y = log(sin x)

^{y}⇒ log y = y log(sin x)

**Question 6. If **** , prove that **

**Solution:**

We have,

⇒ y = (tan x)

^{y}Taking log on both sides,

log y = log(tan x)

^{y}⇒ log y = y log tan x

Differentiating with respect to x using chain rule,

Now,

**Question 7. If **** , prove that **

**Solution:**

We have,

⇒ y = u + v + w

where

Now,

Taking log on both sides,

Differentiating with respect to x,

Taking log on both sides,

Taking log on both sides

Using equation in equation (i), we get

**Question 8. If ****, Prove that **

**Solution:**

We have,

⇒ y = (cos x)

^{y}Taking log on both sides,

log y = log(cos x)

^{y}⇒ log y = y log (cos x)

Differentiating with respect to x using chain rule,

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