How do you differentiate #y=ssqrt(1-s^2)+arccoss#?

Question

Elijah Abram · Accepted Answer

#dy/(ds) = (-2s^2)/sqrt(1-s^2)#

Start by splitting into two separate bits and differentiating them one at a time (sorry for sticking in random letters all over the place - I have tried to avoid #y#, where a #y# is misleading!).

Suppose #k=s*sqrt(1-s^2)# and #m=arccos(s)# and #u=sqrt(1-s^2)#

First #d/(ds)(s*sqrt(1-s^2))#, is determined using the chain rule and product rule.

#(dk)/(ds)= s*d/(ds)(u(s)) + u*d/(ds)(s)#

Set #t=1-s^2# and #u=t^(-1/2)# then apply the chain rule #(du)/(ds) = (du)/(dt) * (dt)/(ds)#.

Hence, #dy/(ds) = -2s * 1/2*(1-s^2)^(-1/2)# This simplifies down to #(du)/(ds) = -s/(sqrt(1-s^2)#.

Next, to apply the product rule to find the overall derivative.

We have #s# times that derivative, and the derivative of #s# times the undifferentiated expression:

#d/(ds)(s*sqrt(1-s^2)) = (-s^2)/sqrt(1-s^2) + sqrt(1-s^2)#.

When you find a common denominator by algebraic manipulation, hence:

#(dk)/(ds) = -(2s^2-1)/sqrt(1-s^2)#.

Then, there is the differentiation of the #arccos# function.

Apply #cos# to both sides to undo the the effect of the inverse #cos#. Hence #m=arccos(s)# becomes #s=cos(m)#.

Using the knowledge that #d/dx(cos(x)) = -sin(x)#, we can say that #(ds)/(dm)= -sin(m)#

Next, to remove the necessity of the #sin# function in there, we can use the Pythagorean identity to find the derivative in terms of #s# only.

Since #cos^2(theta) + sin^2(theta) = 1#, it follows that #sin(theta) = sqrt(1-cos^2(theta))#.

Hence, #(ds)/(dm) = -sqrt(1-s^2#, because #s=cos(m)#.

We know from the chain rule that #dy/dx = 1/(dx/dy)#, so finally #d/(ds)(arccos(s)) = -1/sqrt(1-s^2)#.

The simplification of the two together is quite delightful, as the already have the same denominator - so you can just add!

Finally, we have #dy/(ds) = -((2s^2) / sqrt(1-s^2))#.

Sorry for all the confusion here, I hope it helps!

Evelyn Agard · Answer

undefined To differentiate $y = s\sqrt{1 - s^2} + \arccos(s)$, apply the chain rule and the derivatives of square root and arccosine functions.

$y' = \frac{d}{ds}[s\sqrt{1 - s^2}] + \frac{d}{ds}[\arccos(s)]$  
$y' = \sqrt{1 - s^2} + \frac{-1}{\sqrt{1 - s^2}}$  
$y' = \frac{\sqrt{1 - s^2} - 1}{\sqrt{1 - s^2}}$