h(x)= (e^x)^(1/2)

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h(x)= (e^x)^(1/2)

h'(x)=?

h'(x) = e^x/2 / 2

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Find the derivative of the following via implicit differentiation:
d/dx(h(x)) = d/dx(sqrt(e^x))
The derivative of h(x) is h'(x):
h'(x) = d/dx(sqrt(e^x))
Using the chain rule, d/dx(sqrt(e^x)) = ( dsqrt(u))/( du) ( du)/( dx), where u = e^x and ( d)/( du)(sqrt(u)) = 1/(2 sqrt(u)):
h'(x) = (d/dx(e^x))/(2 sqrt(e^x))
The derivative of e^x is e^x:
h'(x) = e^x/(2 sqrt(e^x))
Simplify the expression:
h'(x) = sqrt(e^x)/2
Expand the left hand side:
Answer: |  h'(x) = sqrt(e^x)/2

h(x)= (e^x)^(1/2)

h'(x)=?

$\begin{array}{rcll} h(x) &=& (e^x)^{\frac12} \\ h'(x) &=& \frac12 \cdot (e^x)^{(\frac12-1)} \cdot e^x \\ h'(x) &=& \frac12 \cdot (e^x)^{(\frac12-1)} \cdot (e^x)^1 \\ h'(x) &=& \frac12 \cdot (e^x)^{(\frac12-1+1)} \\ h'(x) &=& \frac12 \cdot (e^x)^{\frac12} \\ h'(x) &=& \frac12 \cdot e^{ (\frac{x}{2}) } \\ h'(x) &=& \frac { e^{ (\frac{x}{2})} }{2} \end{array}$