If and find the values of at and

OR

If prove that

Given: x = a sin 2t(1 + cos 2t) and y = b cos 2t(1 – cos 2t)

x = a sin 2t(1 + cos 2t)

Differentiating both sides with respect to t:

y = b cos 2t(1 – cos 2t)

Differentiating both sides with respect to t:

Now,

OR

Given: y = x^x

To prove:

y = x^x

Taking log both sides:

⇒ log y = log x^x

⇒ log y = x log x

{∵ log (a^b) = b log a}

Differentiating both sides with respect to x:

Differentiating both sides again with respect to x:

Let c = x^x

Taking log both the sides:

⇒ log c= log x^x

⇒ log c = x log x

{∵ log x^a = alog x}

Differentiating with respect to x:

Put the value of c = x^x

From (i):

Hence Proved