Find the derivative of the function and evaluate the derivative at the given value of a.

Question

h(x)=x^{square root of x;a=16}

 

d/dx=x^{square root of x}=x^{square root of x}(2+lnx/2square root of x)

 

Evaluate the derivative at the given value of a.

 

h'(a)=?

 

 

 

Answer 1

h(x) = x^√x

 

When you get a problem like this, try taking the natural log of both sides, differentiate using the chain rule, and then solve for h'(x).

 

ln[h(x)] = √x lnx

 

Take d/dx of both sides.

 

h'(x)/h(x) = (1/2)x^-1/2lnx + √x (1/x)

 

Solve for h'.

 

h'(x) = h(x) [(1/2)x^-1/2lnx + √x (1/x)] = x^√x [(1/2)x^-1/2lnx + √x (1/x)]

 

h'(a) = a^√a [(1/2)a^-1/2lna + √a (1/a)]

 

You can simplify this further if you wish, and plug in a = 16.  You should be able to finish from here.