# 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 $$\frac{\partial z}{\partial y}$$.