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.

Exercise · ZW-MULTIVARIABLE-CHAIN-RULE-009 · ★★★ Advanced

Let \(z=f(x,y)\) with \(f_x(1,1)=2\), \(f_y(1,1)=3\), \(x=t^2\), \(y=t^3\). Compute \(\dfrac{dz}{dt}\) at \(t=1\).

Choose one option.

Show answer

B. \(13\)

Explanation

At \(t=1\) we are at \((1,1)\), with \(x'=2t=2\) and \(y'=3t^2=3\): \(\frac{dz}{dt}=2\cdot2+3\cdot3=13\).

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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.

\[\frac{dz}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}\]

Let \(z=f(x,y)\) with \(f_x(1,1)=2\), \(f_y(1,1)=3\), \(x=t^2\), \(y=t^3\). Compute \(\dfrac{dz}{dt}\) at \(t=1\).

Explanation

Review the concept: Chain rule in several variables

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