Question

f(x)=x^0.5⋅e^( x^2−x)⋅ (x+1) ^( 2/3 ) how to differentiate??

Answer 1

\[\LARGE f(x)\quad=\quad x^{0.5}\cdot e^{(x^2-x)}\cdot (x+1)^{2/3}\]So we need to find the derivative of this?

Answer 2

You have a couple of options:
~ The way your teacher or whoever would probably like you to solve this is by logarithmic differentiation. We take the natural log of each side, the we can use rules of logs to pull the pieces apart making it easier to differentiate.
~ You could alternatively use the product rule. it ends up following the same pattern that you saw with the product of 2 functions.

\[\Large\bf \left(uv\right)'\quad=\quad \color{#CC0033}{u'}v+u\color{#CC0033}{v'}\]\[\Large\bf\left(uvw\right)'\quad=\quad \color{#CC0033}{u'}vw+u\color{#CC0033}{v'}w+uv\color{#CC0033}{w'}\]