If \(f\colon\, U \subset {\mathbb R}^{\it n}\rightarrow {\mathbb R}\) is differentiable, prove that \({\bf x}\mapsto f^2({\bf x})+2f({\bf x})\) is differentiable as well, and compute its derivative in terms of \({\bf D}\! f({\bf x})\).
Prove that the following functions are differentiable, and find their derivatives at an arbitrary point:
Verify the first special case of the chain rule for the composition \(f\circ {\bf c}\) in each of the cases:
What is the velocity vector for each path \({\bf c}(t)\) in Exercise 3? [The solution to part (b) only is in the Study Guide to this text.]
Let \(f\colon\, {\mathbb R}^3\rightarrow {\mathbb R}\) and \(g\colon\, {\mathbb R}^3\rightarrow {\mathbb R}\) be differentiable. Prove that \[ \nabla (fg)=f \nabla \! g+g \nabla \! f. \]
Let \(f\colon\, {\mathbb R}^3\rightarrow {\mathbb R}\) be differentiable. Making the substitution \[ x=\rho \cos \theta \sin \phi,\quad y=\rho\sin \theta \sin\phi,\quad z=\rho \cos \phi \] (spherical coordinates) into \(f(x,y,z)\), compute \(\partial f/\partial \rho,\partial f/\partial \theta,\) and \(\partial f/\partial \phi\) in terms of \(\partial f/\partial x, \partial f/\partial y, \hbox{and } \partial f/\partial z\).
Let \(f(u,v)=(\tan\, (u-1)-e^v,u^2-v^2)\) and \(g(x,y)=(e^{x-y},x-y)\). Calculate \(f\circ g\) and \({\bf D}(f\circ g)(1,1)\).
Let \(f(u,v,w)=(e^{u-w},\cos\,(v\,+\,u)\,+ \sin\,(u\,+\,v\,+w))\) and \(g(x,y)= (e^x,\cos\, (y-x),e^{-y})\). Calculate \(f\circ g\) and \({\bf D}(f\circ g)(0,0)\).
Find \((\partial /\partial s)(f\circ T)(1,0)\), where \(f(u,v)=\cos\, u\sin\, v\) and \(T{:}\, {\mathbb R}^2\rightarrow {\mathbb R}^2\) is defined by \(T(s,t)=(\cos \,(t^2s), \log\,\sqrt{1+s^2})\).
Suppose that the temperature at the point \((x,y,z)\) in space is \(T(x,y,z)=x^2+y^2+z^2\). Let a particle follow the right-circular helix \({\sigma}(t)=(\cos t,\sin t,t)\) and let \(T(t)\) be its temperature at time \(t\).
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Let \(f(x, y, z)=(3y+2, x^2+y^2, x+z^2)\). Let \(\textbf{c}(t)=(\cos(t), \sin(t), t)\).
Let \(h\colon \mathbb{R}^3 \to \mathbb{R}^5\) and \(g\colon \mathbb{R}^2 \to \mathbb{R}^3\) be given by \(h(x,y,z)= (xyz, e^{xz}, x\sin (y), \frac{-9}{x}, 17)\) and \(g(u,v)=(v^2 + 2u, \pi, 2\sqrt{u})\). Find \(\textbf{D}(h \circ g)(1,1)\).
Suppose that a duck is swimming in the circle \(x=\cos t,y=\sin t\) and that the water temperature is given by the formula \(T=x^2e^y-xy^3\). Find \(dT/dt\), the rate of change in temperature the duck might feel: (a) by the chain rule; (b) by expressing \(T\) in terms of \(t\) and differentiating.
Let \(f\colon\, {\mathbb R}^n\to {\mathbb R}^m\) be a linear mapping so that (by Exercise 28, Section 2.4) \({\bf D}\! f({\bf x})\) is the matrix of \(f\). Check the validity of the chain rule directly for linear mappings.
Let \(f\colon\, {\mathbb R}^2\to {\mathbb R}^2;(x,y)\mapsto (e^{x+y},e^{x-y})\). Let \({\bf c}(t)\) be a path with \({\bf c}(0)=(0,0)\) and \({\bf c}'(0)=(1,1)\). What is the tangent vector to the image of \({\bf c}(t)\) under \(f\) at \(t=0\)?
Let \(f(x,y)=1/\sqrt{x^2+y^2}\). Compute \(\nabla \! f(x,y)\).
Write out the chain rule for each of the following functions and justify your answer in each case using Theorem 11.
Verify the chain rule for \(\partial h/\partial x\), where \(h(x,y)=f(u(x,y),v(x,y))\) and \[ f(u,v)=\frac{u^2+v^2}{u^2-v^2}, u(x,y)=e^{-x-y}, v(x,y)=e^{xy}. \]
(c) Let \(y\) be defined implicitly by \[ x^2+y^3+e^y=0. \]
Compute \(dy/dx\) in terms of \(x\) and \(y\).
Thermodynamics textsfootnote # use the relationship \[ \Big(\frac{\partial y}{\partial x}\Big) \Big(\frac{\partial z}{\partial y}\Big) \Big(\frac{\partial x}{\partial z}\Big)=-1. \]
Explain the meaning of this equation and prove that it is true. [HINT: Start with a relationship \(F(x,y,z)=0\) that defines \(x=f(y,z),y=g(x,z)\), and \(z=h(x,y)\) and differentiate implicitly.]
Dieterici’s equation of state for a gas is \[ P(V-b)e^{a/RVT}=RT, \] where \(a,b\), and \(R\) are constants. Regard volume \(V\) as a function of temperature \(T\) and pressure \(P\) and prove that \[ \frac{\partial V}{\partial T}=\Big(R+\frac{a}{TV}\Big)\Big/ \Big(\frac{RT}{V-b}-\frac{a}{V^2}\Big). \]
This exercise gives another example of the fact that the chain rule is not applicable if \(f\) is not differentiable. Consider the function \[ f(x,y)=\left\{ \begin{array}{lc} \displaystyle \frac{xy^2}{x^2+y^2}& (x,y)\not=(0,0)\\[10pt] 0 & (x,y)=(0,0). \end{array}\right. \]
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Show that
Prove that if \(f\colon\, U\subset {\mathbb R}^n\to {\mathbb R}\) is differentiable at \({\bf x}_0 \in U\), there is a neighborhood \(V\) of \({\bf 0}\in {\mathbb R}^n\) and a function \(R_1\colon\, V\to {\mathbb R}\) such that for all \({\bf h}\in V\), we have \({\bf x}_0+{\bf h}\in U\), \[ f({\bf x}_0+{\bf h})=f({\bf x}_0)+[{\bf D}\! f({\bf x}_0)]{\bf h}+R_1({\bf h}) \] and \[ \frac{R_1({\bf h})}{ \| {\bf h} \| }\to 0 {\rm as} {\bf h}\to {\bf 0}. \]
Suppose \({\bf x}_0\in {\mathbb R}^n\) and \(0\leq r_1<r_2\). Show that there is a \(C^1\) function \(f\colon\, {\mathbb R}^n\to {\mathbb R}\) such that \(f({\bf x})=0\) for \(\| {\bf x}-{\bf x}_0 \| \geq r_2; 0 <f({\bf x})<1\) for \(r_1< \| {\bf x}-{\bf x}_0 \| <r_2\); and \(f({\bf x})=1\) for \(\| {\bf x}-{\bf x}_0 \| \leq r_1\). [HINT: Apply a cubic polynomial with \(g(r_1^2)=1\) and \(g(r^2_2)= g'(r^2_2)= g'(r_1^2)=0\) to \(\| {\bf x}-{\bf x}_0 \| ^2\) when \(r_1< \| {\bf x}-{\bf x}_0 \| <r_2.\)]
Find a \(C^1\) mapping \(f\colon\, {\mathbb R}^3\to {\mathbb R}^3\) that takes the vector \({\bf i}+{\bf j}+{\bf k}\) emanating from the origin to \({\bf i}-{\bf j}\) emanating from \((1, 1, 0)\) and takes \({\bf k}\) emanating from \((1, 1, 0)\) to \({\bf k}-{\bf i}\) emanating from the origin.
What is wrong with the following argument? Suppose \(w=f(x,y,z)\) and \(z=g(x,y)\). By the chain rule, \[ \frac{\partial w}{\partial x}=\frac{\partial w}{\partial x}\frac{\partial x}{\partial x}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial x}+\frac{\partial w}{\partial z}\frac{\partial z}{\partial x}=\frac{\partial w}{\partial x}+\frac{\partial w}{\partial z}\frac{\partial z}{\partial x}. \]
Hence, \(0=(\partial w/\partial z)(\partial z/\partial x)\), and so \(\partial w/\partial z=0\) or \(\partial z/\partial x=0\), which is, in general, absurd.
Prove rules (iii) and (iv) of Theorem 10. (HINT: Use the same addition and subtraction tricks as in the one-variable case and Theorem 8.)
Show that \(h\colon\, {\mathbb R}^n\to {\mathbb R}^m\) is differentiable if and only if each of the \(m\) components \(h_i\colon\, {\mathbb R}^n\to {\mathbb R}\) is differentiable. (HINT: Use the coordinate projection function and the chain rule for one implication and consider \begin{eqnarray*} && \Big[\frac{\| h({\bf x})\,-\,h({\bf x}_0) -{\bf D}h({\bf x}_0) ({\bf x}\,-\,{\bf x}_0) \| }{ \| {\bf x}\,-\,{\bf x}_0 \|}\Big]^2\\[4pt] &&=\frac{\sum^m_{i=1}[h_i({\bf x})-h_i({\bf x}_0){\bf D} h_i({\bf x}_0)({\bf x}\,-\,{\bf x}_0)]^2}{\|{\bf x}\,-\,{\bf x}_0 \|^2} \end{eqnarray*} to obtain the other.)
Use the chain rule and differentiation under the integral sign, namely, \[ \frac{d}{dx}\int^b_a f(x,y)\,dy= \int^b_a\frac{\partial f}{\partial x}(x,y)\,dy, \] to show that \[ \frac{d}{d x}\int ^x_0f(x,y)\,dy=f(x,x)+\int^x_0\frac{\partial f}{\partial x}(x,y)\,dy. \]
For what integers \(p>0\) is \[ f(x)=\Big\{ \begin{array}{lc} x^p\sin\, (1/x) & x\not=0\\[1pt] 0& x=0 \end{array} \] differentiable? For what \(p\) is the derivative continuous?
Suppose \(f\colon\, {\mathbb R}^n\to {\mathbb R}\) and \(g\colon\, {\mathbb R}^n\to {\mathbb R}^m\) are differentiable. Show that the product function \(h({\bf x})=f({\bf x})g({\bf x})\) from \({\mathbb R}^n\) to \({\mathbb R}^m\) is differentiable and that if \({\bf x}_0\) and \({\bf y}\) are in \({\mathbb R}^n\), then \([{\bf D}h({\bf x}_0)]{\bf y}= f({\bf x}_0)\{[{\bf D}g({\bf x}_0)]{\bf y}\}+\{[{\bf D}\! f({\bf x}_0)]{\bf y}\}g({\bf x}_0)\).
Let \(g(u, v)=(e^u, u+\sin v)\) and \(f(x, y, z)=(xy, yz)\). Compute \(\textbf{D}(g \circ f)\) at (0, 1, 0) using the chain rule.
Let \(f\colon\mathbb{R}^4 \to \mathbb{R}\) and \(\textbf{c}(t)\colon \mathbb{R} \to \mathbb{R}^4\). Suppose \(\nabla f(1, 1, \pi, e^6)=(0, 1, 3, -7), \ \textbf{c}(\pi)= (1, 1, \pi, e^6),\) and \(\textbf{c}'(\pi)= (19, 11, 0, 1)\). Find \(\displaystyle \frac{d(f \circ \textbf{c})}{dt}\) when \(t=\pi\).
Suppose \(f\colon \mathbb{R}^n \to \mathbb{R}^m\) and \(g\colon \mathbb{R}^p \to \mathbb{R}^q\).
If \(z=f(x-y)\), use the chain rule to show that \(\displaystyle \frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}=0\).
Let \(w=x^2+y^2+z^2, \ x=uv, \ y=u \cos v, z=u \sin v\). Use the chain rule to find \(\displaystyle \frac{\partial w}{\partial u}\) when\((u, v)=(1, 0)\).