Exercise · ZW-MULTIVARIABLE-CHAIN-RULE-005 · ★★☆ Intermediate
Let \(z=\ln(x^2+y^2)\) with \(x=t\), \(y=2t\). Compute \(\dfrac{dz}{dt}\) at \(t=1\).
Choose one option.
Show answer
- B. \(2\)
Explanation
\(\frac{dz}{dt}=\frac{2x\cdot1+2y\cdot2}{x^2+y^2}\). At \(t=1\): \(\frac{2+8}{1+4}=2\). (Check: \(z=\ln5+2\ln t\).)
Review the concept: Chain rule in several variables
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-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)?
- ZW-MULTIVARIABLE-CHAIN-RULE-009 Let z=f(x,y) with f_x(1,1)=2, f_y(1,1)=3, x=t^2, y=t^3. Compute (dz)/(dt) at t=1.
← ZW-MULTIVARIABLE-CHAIN-RULE-004 ZW-MULTIVARIABLE-CHAIN-RULE-006 →