Question

I don’t get either of it since I didn’t get the chance to revise it, but the homework is due very soon, so I’ve been left with no choice :(

Answer 1

Given the function:

`x^{(1)/(2)}\ln (3x)`

We will differentiate with respect to x

For the given function, we will use the product rule

`(f\cdot g)^(\prime)=f^(\prime)\cdot g+f\cdot g^(\prime)`

we have two functions:

`\begin{gathered} (x^{(1)/(2)})^(\prime)=\frac{1}{2x^{(1)/(2)}} \\ \ln (3x)^(\prime)=(1)/(3x)\cdot3=(1)/(x) \end{gathered}`

So, the first derivative for the given function will be:

`\begin{gathered} (x^{(1)/(2)}\ln \lbrack3x\rbrack)^(\prime)=\frac{1}{2x^{(1)/(2)}}\ln (3x)+x^{(1)/(2)}\cdot(1)/(x) \\ \\ =\frac{1}{2x^{(1)/(2)}}\ln (3x)+\frac{2x^{(1)/(2)}}{2x^{(1)/(2)}x^{(1)/(2)}} \\ \\ =\frac{1}{2x^{(1)/(2)}}\ln (3x)+\frac{2}{2x^{(1)/(2)}} \\ \\ =\frac{1}{2x^{(1)/(2)}}(\ln (3x)+2) \end{gathered}`