How do I find the derivative of # y=xsin^(-1)x + sqrt(1-x^2)#?
1 Answer
Explanation:
You can differentiate this function by using the product rule and the chain rule, provided that you know that
#d/dx(arcsinx) = 1/(sqrt(1-x^2)#
So, your function can be written like this
#y = x * arcsinx + sqrt(1-x^2)#
You will use the product rule to differentiate
#d/dx(x * arcsinx) = [d/dx(x)] * arcsinx + x * d/dx(arcsinx)#
#d/dx(x * arcsinx) = 1 * arcsinx + x * 1/(sqrt(1-x^2)#
and
#d/dx(sqrt(u)) = d/(du)sqrt(u) * d/dx(u)#
#d/dx(sqrt(u)) = 1/2 * 1/sqrt(u) * d/dx(1-x^2)#
#d/dx(sqrt(1-x^2)) = 1/color(red)(cancel(color(black)(2))) * 1/sqrt(1-x^2) * (-color(red)(cancel(color(black)(2)))x)#
#d/dx(sqrt(1-x^2)) = -x/sqrt(1-x^2)#
The derivative of
#d/dx(y) = d/dx(x * arcsinx) + d/dx(sqrt(1-x^2))#
#y^' = arcsinx + x/sqrt(1-x^2) + (-x/sqrt(1-x^2))#
#y^' = arcsinx + color(red)(cancel(color(black)(x/sqrt(1-x^2)))) -color(red)(cancel(color(black)(x/sqrt(1-x^2))))#
#y^' = color(green)(arcsinx)#