Question

can this be differentiated? And if so how.
y = sqrt(x) - (1/2)^x

Answer 1

For any positive number \( a \), suppose \( y = a^x \). Then
\[ \ln y = x \ln a \] and hence
\[ \frac{1}{y}y' = \ln a \]
\[ y' = y \ln a = (\ln a)a^x \]

Therefore for the function \( f(x) = (1/2)^x \), the derivative is
\[ \frac{df}{dx} = \ln(1/2).(1/2)^x = -(\ln 2) 2^{-x} \]

Answer 2

Make sense?