Question

Question #765ea

Answer 1

`"The Reqd. Derivative="(sqrte)^sqrtx/(4sqrtx).`

Explanation:

Let `y=(sqrte)^(sqrtx)=(e^(1/2))^sqrtx=e^(1/2sqrtx)=e^t,` say, where, `t=1/2sqrtx`.

Thus, `y=e^t, t=1/2sqrtx=1/2x^(1/2),` whivh means that,

`y" is a function of "t, and, t" of "x.`

In such cases, we use the Chain Rule to find `dy/dx," i.e. to say,"`

`dy/dx=dy/dt dt/dx.....................(star)`

Now, `because, y=e^t, dy/dt=e^t,`

`and, t=1/2x^(1/2), dt/dx=1/2{1/2x^(1/2-1)}=1/4x^(-1/2)=1/(4sqrtx).`

Utilising all these in `(star), dy/dx=e^t/(4sqrtx)`

Reverting back from `t" to "x`, we finally get,

`"The Reqd. Derivative="e^(1/2sqrtx)/(4sqrtx)=(sqrte)^sqrtx/(4sqrtx).`

Enjoy Maths.!