How do I use the chain rule to find the derivative of

Question

square root(1-x^2)

also, are there any general hints or tips for determining when the chain rule and product or quotient rule should be used?? i'm having trouble discerning when both the chain rule and either the quotient or product rules are necessary to solve the problem.

thank you.

Answer 1

This is often a toss-up

generally if you have a single power, such as yours, use the chain rule.

I you see a multiplication (hence product) use the product rule, etc.

at times you have to use one rule while applying another.
e.g. y = (3x+5)(2x-5)^5
is primarily a product rule, but while you are doing that you will have to use the chain rule for the power.

for your question first change the root sign to an exponent of 1/2

y = (1-x^2)^(1/2)

dy/dx = 1/2(1-x^2)^(-1/2)(2x)
= x(1-x^2)^(-1/2) or x/√(1-x^2)