Caluclus: Table of contents

Chain Rule (Multivariable Calculus)

Chain rule

Theorem (Chain rule) Assume that \(x,y:\mathbb R\to\mathbb R\) are differentiable at point \(t_0\). Assume that \(f:\mathbb R\times\mathbb R\to\mathbb R\) is differentaible at point \((x(t_0),y(t_0))\) with continuous partial derivatives. Then the function \(g(t)=f(x(t),y(t))\) is differentiable at the point \((x(t_0),y(t_0))\) and satisfies \[\frac{d g}{dt}(t_0)=\frac{\partial f}{\partial x}(x(t_0),y(t_0))\cdot \frac{dx}{d t}(t_0)+ \frac{\partial f}{\partial y}(x(t_0),y(t_0))\cdot \frac{dy}{d t}(t_0).\]

Implicit function theorem

Theorem (Implicit function theorem) Let \(F:\mathbb R^3\to\mathbb R\) be a continuously differentiable function such that \(\frac{\partial F}{\partial z}(x_0,y_0,z_0)\neq 0\). Then the implicit equation \(F(x,y,z)=F(x_0,y_0,z_0)\) has a solution \(z=f(x,y)\) in a neighborhood of the point \((x_0,y_0)\) and the partial derivatives of \(f\) satisfy: \[\frac{\partial f}{\partial x}(x_0,y_0)=-\frac{\frac{\partial F}{\partial x}(x_0,y_0)}{\frac{\partial F}{\partial z}(x_0,y_0)}\quad\quad\quad \mbox{and}\quad\quad\quad \frac{\partial f}{\partial y}(x_0,y_0)=-\frac{\frac{\partial F}{\partial y}(x_0,y_0)}{\frac{\partial F}{\partial z}(x_0,y_0)}. \]

Practice problems

Problem 1. Assume that \(f(x,y,z)=x^2+y^3+z^4\), and that \(x(s,t)=s\cos t\), \(y(s,t)=st\), \(z(s,t)=s^2\). Define the function \[g(s,t)=f\left(x(s,t),y(s,t),z(s,t)\right).\] Determine the value of \(\frac{\partial g}{\partial t}\left(2,\pi\right)\).

Problem 2. If \(z(x,y)=\ln(x+3y)\), \(x(t)=t^2\), \(y(t)=\sin t\), find \(\frac{dz}{dt}\).

Problem 3. Assume that \(x(s,t)=s^2+t^2\) and \(y(s,t)=2st\). Assume that \(f:\mathbb R\times\mathbb R\to\mathbb R\) is a three time differentiable function and \(g(s,t)=f(x(s,t),y(s,t))\). If \(\frac{\partial f}{\partial x}(10,6)=7\), \(\frac{\partial f}{\partial y}(10,6)=8\), \(\frac{\partial^2 f}{\partial x^2}(10,6)=1\), \(\frac{\partial^2f}{\partial y^2}(10,6)=2\), and \(\frac{\partial^2f}{\partial x\partial y}(10,6)=3\), find \[\frac{\partial^2g}{\partial s\partial t}(1,3).\]

Problem 4. Assume that \(a\) is a positive real number and that \(u:\mathbb R\to\mathbb R\) is a twice differentiable function. Define \(f(x,t)=u(x+at)+u(x-at)\). Determine the value of \[\dfrac{\mbox{ }\frac{\partial^2 f}{\partial x^2}\mbox{ }}{\frac{\partial^2 f}{\partial t^2}}.\]

Problem 5. If the function \(z=z(x,y)\) is defined implicitly as \(xyz=\cos(x+y+z)\), find implicit equations for the functions \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\).