# Partial Derivatives and Appropriate Chain Rule

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Find partial deriv's w/r and w/theta using appropriate chain rule for :

w=the square root of (25-5x^2-5y^2), where x=r cos theta, y= r sin theta.

https://brainmass.com/math/derivatives/partial-derivatives-appropriate-chain-rule-9015

## SOLUTION This solution is **FREE** courtesy of BrainMass!

w = (25 - 5r^2(cos^2 (theta) + sin^2(theta)))^1/2 or

w = (25 - 5r^2)^1/2 ( as (cos^2 (theta) + sin^2(theta) = 1)

therefore dw/dr = 1/2*(25 - 5r^2)^-1/2*-10r or

dw/dr = -5r*(25 - 5r^2)^-1/2

dw/dtheta = dw/dx* dx/dtheta + dw/dy * dy/dtheta

= 1/2((25 - 5x^2 - 5y^2)^-1/2)*-10x*(-r sin(theta))+ 1/2((25 - 5x^2 - 5y^2)^-1/2)*-10y* (rcos(theta))

Hope it is clear.

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