Question

How do you find the derivative of the function `(3x-2)/(2x+1)^(1/2)`?

Answer 1

`(3x+5)/(2x+1)^(3/2)`

Explanation:

`"differentiate using the "color(blue)"quotient rule"`

`"given "f(x)=(g(x))/(h(x))" then"`

`f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larr" quotient rule"`

`g(x)=3x-2rArrg'(x)=3`

`"differentiate "(2x+1)^(1/2)" using the "color(blue)"chain rule"`

`h(x)=(2x+1)^(1/2)rArrh'(x)=1/2(2x+1)^(-1/2)xx2`

`color(white)(xxxxxxxxxxxxxxxxxx)=(2x+1)^(-1/2)`

`rArrf'(x)=((2x+1)^(1/2).3-(3x-2)(2x+1)^(-1/2))/((2x+1)^(1/2))^2`

`color(white)(rArrf'(x))=((2x+1)^(-1/2)(3(2x+1)-(3x-2)))/(2x+1)`

`color(white)(rArrf'(x))=(3x+5)/(2x+1)^(3/2)`