How do you find the derivative of arcsinx?

to find the derivative, we can first make this easier to visualise. Say a second variable, y, is included to make this become y = arcsinx. now if we sin both sides, we get siny = x. we already know the derivative of siny is cosy, but of course we are trying to find d/dx. if we simply interchange siny with cosy we are actually finding d/dy. there are 2 ways to go from this point. we can use the "chain rule", which states that d/dy * dy/dx = d/dx. as we know that d/dy siny = cosy, we now get d/dx siny = cosy X dy/dx. doing the same to both sides, d/dx x = 1. we now have cosy dy/dx = 1, giving us dy/dx = 1/cosy. however we do not want our added variable to be included in our answer, so we need to get rid of the cosy and replace it with a function of xs. How can we do this? we can use the substitution sin²y + cos²y = 1. as x = siny from before, this becomes x² + cos²y = 1. rearranging this gives cosy = (1-x²)^1/2 and d/dx arcsinx = 1/(1-x²)^1/2.