I'm having a little trouble with this one: (1+xy)^3 = 3y +5
P parsons9 New member Joined Sep 1, 2008 Messages 6 Dec 8, 2008 #1 I'm having a little trouble with this one: (1+xy)^3 = 3y +5
S soroban Elite Member Joined Jan 28, 2005 Messages 5,586 Dec 8, 2008 #2 Re: implicit differientiation Hello, parsons9! Did you forget the Product Rule? Differentiate implicitly: .\(\displaystyle (1+xy)^3 \:= \:3y +5\) Click to expand... \(\displaystyle \text{We have: }\;3(1+xy)^2\cdot\left(x\!\cdot\!\tfrac{dy}{dx} + 1\!\cdot\!y\right) \;=\;3\,\tfrac{dy}{dx}\) . . . . . \(\displaystyle 3x(1+xy)^2\,\tfrac{dy}{dx} + 3y(1+xy)^2 \;=\;3\,\tfrac{dy}{dx}\) . . . . . . . . . \(\displaystyle 3x(1+xy)^2\,\tfrac{dy}{dx} - 3\,\tfrac{dy}{dx} \;=\;-3y(1+xy)^2\) \(\displaystyle \text{Factor: }\;\;3\bigg[x(1+xy)^2 - 3\bigg]\,\tfrac{dy}{dx} \;=\;-3y(1+xy)^2\) \(\displaystyle \text{Therefore: }\:\frac{dy}{dx} \;=\;\frac{-3y(1+xy)^2}{3[x(1+xy)^2-3]} \;=\;\frac{y(1+xy)^2}{3-x(1+xy)^2}\) Edit: Corrected my omission . . .
Re: implicit differientiation Hello, parsons9! Did you forget the Product Rule? Differentiate implicitly: .\(\displaystyle (1+xy)^3 \:= \:3y +5\) Click to expand... \(\displaystyle \text{We have: }\;3(1+xy)^2\cdot\left(x\!\cdot\!\tfrac{dy}{dx} + 1\!\cdot\!y\right) \;=\;3\,\tfrac{dy}{dx}\) . . . . . \(\displaystyle 3x(1+xy)^2\,\tfrac{dy}{dx} + 3y(1+xy)^2 \;=\;3\,\tfrac{dy}{dx}\) . . . . . . . . . \(\displaystyle 3x(1+xy)^2\,\tfrac{dy}{dx} - 3\,\tfrac{dy}{dx} \;=\;-3y(1+xy)^2\) \(\displaystyle \text{Factor: }\;\;3\bigg[x(1+xy)^2 - 3\bigg]\,\tfrac{dy}{dx} \;=\;-3y(1+xy)^2\) \(\displaystyle \text{Therefore: }\:\frac{dy}{dx} \;=\;\frac{-3y(1+xy)^2}{3[x(1+xy)^2-3]} \;=\;\frac{y(1+xy)^2}{3-x(1+xy)^2}\) Edit: Corrected my omission . . .
P parsons9 New member Joined Sep 1, 2008 Messages 6 Dec 9, 2008 #3 Re: implicit differientiation when factored should it be: 3[x(1+xy)^2 -1] dy/dx if not, where did the square go and why is it still -3? Thank you for your help! I did forget the product rule.
Re: implicit differientiation when factored should it be: 3[x(1+xy)^2 -1] dy/dx if not, where did the square go and why is it still -3? Thank you for your help! I did forget the product rule.
