Questions: Multivariate chain rule
Before attempting these questions, it is highly recommended that you read Guide: Multivariate chain rule.
Q1
Let \(z=z(x,y)\) be a function where both \(x\) and \(y\) depend on an independent variable \(t\).
For each function given below, use the multivariate chain rule or otherwise to find \(\dfrac{\mathrm{d}z}{\mathrm{d}t}\), expressing your answer in terms of \(t\) only.
1.1. \(\quad \displaystyle z=x^2y\) where \(x=\sin(t)\) and \(y=e^{2t}\).
1.2. \(\quad \displaystyle z = \ln(xy)\) where \(x = t^3\) and \(y = \cos(t)\).
1.3. \(\quad \displaystyle z = x^3 + y^3\) where \(x = \sqrt{t}\) and \(y = t^2 + 1\).
1.4. \(\quad \displaystyle z = e^{xy}\) where \(x = t\) and \(y = \ln(t+1)\).
1.5. \(\quad \displaystyle z = x \tan(y)\) where \(x = \cos(t)\) and \(y = t^2\).
1.6. \(\quad \displaystyle z = x^2 + 3xy + y^3\) where \(x = 2t - 1\) and \(y = 5\sin(t)\).
1.7. \(\quad \displaystyle z = \dfrac{x}{y}\) where \(x = t^2 + 1\) and \(y = t - 2\).
1.8. \(\quad \displaystyle z = \sqrt{x^2 + y^2}\) where \(x = \cos(t)\) and \(y = \sin(t)\).
1.9. \(\quad \displaystyle z = xy^2 + yx^2\) where \(x = e^t\) and \(y = t^3\).
1.10. \(\quad \displaystyle z = \ln(x) + xy\) where \(x = t^2\) and \(y = e^{-t}\).
1.11. \(\quad \displaystyle z = x^2y\) where \(x = 2t\) and \(y = \ln(t)\).
1.12. \(\quad \displaystyle z = x^2 \sin(y)\) where \(x = t^3 + 1\) and \(y = 3t\).
1.13. \(\quad \displaystyle z = \tan^{-1} \left( \dfrac{y}{x} \right)\) where \(x = t\) and \(y = t^2\).
1.14. \(\quad \displaystyle z = xe^y\) where \(x = \ln(t+2)\) and \(y = \sqrt{t}\).
Q2
Let \(z=z(x,y)\) be a function where both \(x\) and \(y\) depend on two independent variables \(s\) and \(t\).
For each function, use the multivariate chain rule to find \(\dfrac{\partial z}{\partial s}\) and \(\dfrac{\partial z}{\partial t}\), expressing your answers in terms of \(s\) and \(t\) only.
2.1. \(\quad \displaystyle z = x^2y\) where \(x=s+t\) and \(y=s^2-t^2\).
2.2. \(\quad \displaystyle z = \ln(x+y)\) where \(x=e^s \cos(t)\) and \(y=e^s \sin(t)\).
2.3. \(\quad \displaystyle z = x^3-3xy\) where \(x=st\) and \(y=s+t\).
2.4. \(\quad \displaystyle z = e^{x+y}\) where \(x=s^2\) and \(y=\ln(t)\).
2.5. \(\quad \displaystyle z = x\sin(y)\) where \(x=s-t^2\) and \(y=st\).
2.6. \(\quad \displaystyle z = x^2 + y^2\) where \(x=\cos(s)\sin(t)\) and \(y=\sin(s)\cos(t)\).
2.7. \(\quad \displaystyle z = xy+x^2\) where \(x=s+t\) and \(y=s-t\).
2.8. \(\quad \displaystyle z = \ln(x) - \ln(y)\) where \(x=s+t\) and \(y=st\).
2.9. \(\quad \displaystyle z = \tan(x+y)\) where \(x=s^2-t\) and \(y=s+t^2\).
2.10. \(\quad \displaystyle z = \tan^{-1} \left( \dfrac{y}{x} \right)\) where \(x=s^2-t^2\) and \(y=2st\).
Q3
Let \(w = w(x_1, \ldots, x_n)\) be a function that depends on variables \(x_1, \ldots, x_n\), where each \(x_i\) is itself a function of \(t_1, \ldots, t_m\).
For each function, write the appropriate form of the multivariate chain rule and find the resulting partial derivatives.
3.1. \(\quad \displaystyle w=x^2+y^2+z^2\) where \(\begin{cases} x=s+t \\ y=s-t \\ z=st \end{cases}\)
3.2. \(\quad \displaystyle w=xy+z\) where \(\begin{cases} x=s+t+u \\ y=st \\ z=t+u \end{cases}\)
3.3. \(\quad \displaystyle w=\sin(xy)+\cos(z)\) where \(\begin{cases} x=s^2 \\ y=t^2 \\ z=s+t \end{cases}\)
3.4. \(\quad \displaystyle w=x^2+y^2\) where \(\begin{cases} x=s+t+u \\ y=s-t+u \end{cases}\)
After attempting the questions above, please click this link to find the answers.
Version history and licensing
v1.0: initial version created 05/25 by Donald Campbell as part of a University of St Andrews VIP project.