Homework 9: Properties of Derivatives

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Graded Problems

(a): Suppose $f:\mathbb {R}^3\rightarrow \mathbb {R}$ is a smooth function. Prove that $f_{xyzx} = f_{zxxy}$ , where these are fourth-order partial derivatives. You are free to use Theorem 4.3 for this problem, but not Theorem 4.5.
(b): Verify the result of (a) for the function $f(x,y,z) = x^3y^2z$ .

Suppose we have differentiable functions $\mathbf {f}:\mathbb {R}^2\rightarrow \mathbb {R}^2$ and $\mathbf {g}:\mathbb {R}^2\rightarrow \mathbb {R}^2$ such that $\begin{align*} \mathbf{g}(1,2) &= (3,0),\\ D\mathbf{f}(x,y) &= \left(\begin{array}{cc} xe^y & x^2y \\ 0 & yx^2+1\end{array}\right),\\ D(\mathbf{f}\circ\mathbf{g})(1,2) &= \left(\begin{array}{cc} 1 & 4 \\ 2 & -1\end{array}\right). \end{align*}$

Find $D\mathbf {g}(1,2)$ .

Professional Problem

(a): Consider differentiable functions $g:\mathbb {R}\rightarrow \mathbb {R}$ and $f:\mathbb {R}^2\rightarrow \mathbb {R}$ such that $f(x,y)=0$ when $y=g(x)$ . Prove that if $\partial f/\partial y\neq 0$ , then $\frac {dg}{dx} = -\frac {\partial f/\partial x}{\partial f/\partial y}.$
(b): Consider the equation $\sin (x) + \cos (y) = 0$ . Make a suitable choice of $f$ , and use the result of (a) to compute $\frac {dy}{dx}$ in terms of $x$ and $y$ .
(c): Use implicit differentiation to verify your answer to (b).

Completion Packet

Consider the functions $\mathbf {f}(x,y) = (x^2+y, xy-\sin (xy))$ and $\mathbf {g}(x,y) = (e^{xy}, x^2y)$ . Compute $D(\mathbf {f} + \mathbf {g})$ in two ways:

(a): By computing the derivative of $\mathbf {f} + \mathbf {g}$
(b): By computing the derivatives of $\mathbf {f}$ and $\mathbf {g}$ , and using the sum rule.

Consider the functions $f(x,y,z) = xy^2z^3$ and $g(x,y,z) = xyz$ .

(a): Verify that $D(fg)(\mathbf {a})=g(\mathbf {a})Df(\mathbf {a})+f(\mathbf {a})Dg(\mathbf {a}).$
(b): Verify that $D(f/g)(\mathbf {a})=\frac {g(\mathbf {a})Df(\mathbf {a})-f(\mathbf {a})Dg(\mathbf {a})}{g(\mathbf {a})^2}.$

Find all second-order partial derivatives of the function $f(x,y) = \ln (xy)$ .

Find all second-order partial derivatives of the function $g(u, v) = e^{u^2+v^2}$ .

Find all second-order partial derivatives of the function $h(x,y,z) = xy^2z^3$ .

Consider the functions $y(s,t) = e^s + e^{st} + e^t$ and $\mathbf {x}(t) = (t, t^2)$ . Compute $D(\mathbf {x}\circ y)$ in two ways:

(a): Determining a formula for the composition $\mathbf {x}\circ y$ , then computing the total derivative.
(b): Computing total derivatives of $\mathbf {x}$ and $y$ , and using the chain rule.

Consider functions $\mathbf {f}:\mathbb {R}^3\rightarrow \mathbb {R}^2$ and $\mathbf {g}:\mathbb {R}^2\rightarrow \mathbb {R}^3$ such that $\mathbf {f}(x,y,z) = (x^2y+e^z, e^x+y)$ and $D\mathbf {g}(x,y) = \left (\begin {array}{ccc}y&x\\0&1\\0&xy\end {array}\right )$ . For each of the given total derivatives, either explain why they do not exist, compute them, or explain what additional information we would need to compute them.

(a): $D(\mathbf {f}\circ \mathbf {g})$
(b): $D(\mathbf {g}\circ \mathbf {f})$

Consider functions $\mathbf {f}:\mathbb {R}^3\rightarrow \mathbb {R}^2$ and $\mathbf {g}:\mathbb {R}^2\rightarrow \mathbb {R}^2$ such that $\mathbf {f}(x,y,z) = (\sin (xyz),\cos (xyz))$ and $D\mathbf {g}(x,y) = \left (\begin {array}{ccc}e^{xy}&0\\0&e^{xy}\end {array}\right )$ . For each of the given total derivatives, either explain why they do not exist, compute them, or explain what additional information we would need to compute them.

(a): $D(\mathbf {f}\circ \mathbf {g})$
(b): $D(\mathbf {g}\circ \mathbf {f})$

Consider the function $\mathbf {g}:\mathbb {R}^3\rightarrow \mathbb {R}^3$ given by $\mathbf {g}(\rho , \theta , \phi ) = (\rho \cos \theta \sin \phi , \rho \sin \theta \sin \phi ,\rho \cos \phi )$ , which changes spherical coordinates to Cartesian coordinates. For any differentiable function $f:\mathbb {R}^3\rightarrow \mathbb {R}$ , compute $\partial f/\partial \rho$ , $\partial f/\partial \theta$ , and $\partial f/\partial \phi$ in terms of $\partial f/\partial x$ , $\partial f/\partial y$ , and $\partial f/\partial z$ .