Derivative of sqrt(x sqrt(x))



• To find the derivative of the function sqrt(x sqrt(x)), we need to apply the chain rule of differentiation.



Step 1: Rewrite the function

Rewrite the function as f(x) = (x(sqrt(x)))^(1/2).



Step 2: Apply the chain rule

Let u = x(sqrt(x)), and v = u^(1/2).

Then, f(x) = v(u).

Now, differentiate f(x) using the chain rule: f'(x) = v'(u) * u'(x).



Step 3: Find the derivatives

Find the derivatives of v(u) and u(x):

- v'(u) = (1/2)u^(-1/2)
- u'(x) = sqrt(x) + (1/2)x^(-1/2)



Step 4: Substitute back

Substitute the derivatives back into f'(x) = v'(u) * u'(x):

f'(x) = (1/2)(x(sqrt(x))^(-1/2) * (sqrt(x) + (1/2)x^(-1/2))



Step 5: Simplify

Simplify the expression to get the final derivative of sqrt(x sqrt(x)).



By following these steps, you can find the derivative of the given function sqrt(x sqrt(x)) using the chain rule of differentiation.