Question

How do you find the derivative of `5x arcsin(x)`?

Answer 1

`(5x)/(sqrt(1-x^2))+5arcsin(x)`

Explanation:

`color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(d/dx(arcsinx)=1/(sqrt(1-x^2)))color(white)(2/2)|)))`

`"let "f(x)=5xarcsin(x)`

differentiate f(x) using the `color(blue)"product rule"`

`"Given "f(x)=g(x).h(x)" then"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(f'(x)=g(x)h'(x)+h(x)g'(x))color(white)(2/2)|)))`

`"here "g(x)=5xrArrg'(x)=5`

`"and "h(x)=arcsin(x)rArrh'(x)=1/(sqrt(1-x^2))`

`rArrf'(x)=5x. 1/(sqrt(1-x^2))+5arcsin(x)`

`rArrf'(x)=(5x)/(sqrt(1-x^2))+5arcsin(x)`