Exercise · ZW-MULTIVARIABLE-CHAIN-RULE-010 · ★★★ Advanced
On the curve \(f(x,y)=x^2+y^2=4\) we have \(y=y(x)\). Using the chain rule on \(0=\dfrac{d}{dx}f(x,y(x))\), find \(\dfrac{dy}{dx}\) at \((\sqrt2,\sqrt2)\).
Choose one option.
A \(1\) B \(0\) C \(-\sqrt2\) D \(-1\) Show answer ← Back to the question
Explanation \(0=f_x\cdot1+f_y\,y'=2x+2y\,y'\), so \(y'=-x/y\). At \((\sqrt2,\sqrt2)\) it equals \(-1\).
The chain rule for \(z=f(x,y)\), with \(x=x(t)\) and \(y=y(t)\), gives the derivative of the composition as the sum of what each intermediate variable contributes: its partial of \(f\) times its own derivative.
\[\frac{dz}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}\]
More practice problems: chain rule in several variables ZW-MULTIVARIABLE-CHAIN-RULE-001 Let z=x^2y with x=t, y=t^2. Compute (dz)/(dt) at t=1.ZW-MULTIVARIABLE-CHAIN-RULE-002 Let z=xy with x=cos t, y=sin t. Compute (dz)/(dt) at t=0.ZW-MULTIVARIABLE-CHAIN-RULE-003 Let z=e^x+2y with x=t^2, y=t. Compute (dz)/(dt) at t=0.ZW-MULTIVARIABLE-CHAIN-RULE-004 Let z=x^2+y^2 with x=2t, y=t+1. Compute (dz)/(dt) at t=1.ZW-MULTIVARIABLE-CHAIN-RULE-005 Let z=ln (x^2+y^2) with x=t, y=2t. Compute (dz)/(dt) at t=1.ZW-MULTIVARIABLE-CHAIN-RULE-006 Let z=x^2+xy with x=s+t, y=st. Compute (∂ z)/(∂ s) at (s,t)=(1,2).ZW-MULTIVARIABLE-CHAIN-RULE-007 With z=x^2+xy, x=s+t, y=st, compute (∂ z)/(∂ t) at (s,t)=(1,2).ZW-MULTIVARIABLE-CHAIN-RULE-008 If w=f(x,y) with x=x(s,t), y=y(s,t), what is (∂ w)/(∂ s)?