The chain rule and logarithms practice question 26

The three terms look alike. In each one, is the variable in the base, in the exponent, or in both? That decides which rule each term needs.

Differentiate term by term. Only $x^{x}$ has the variable in both the base and the exponent; $3^{x}$ has a constant base and $x^{3}$ a constant exponent.

For $y = x^{x}$, take logs ($\log$ is the natural logarithm): $\log y = x\log x$, so $\dfrac{y'}{y} = \log x + 1$. For $3^{x}$, use $\dfrac{d}{dx}a^{x} = a^{x}\log a$.