Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

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38
votes
10answers
3k views

Which of the numbers $1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$ is largest, and how to find out without calculator?

$1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$. I got this question in an Application of Derivatives test. I think log might be used here to compare the values, but even then the values ...
2
votes
2answers
32 views

Continuously changing Dimensions of a Rectangle

The dimension of a rectangle are continousl changing. The width increases at a rate of 3 in/s while the length decreases at the rate of 2 in/s. At one instant the rectangle is a 20-in square. How ...
0
votes
1answer
21 views

T o' clock at $T^2$ Miles

A bicycle travels along a straight road. At t o' clock it is $t^2$ miles from the end of the road. Compute its average velocity from 1:00 to 4:00; and instantaneous velocity at 3:00. Answer for first ...
1
vote
1answer
20 views

Minimum Surface Area of a Closed Cylindrical Container

This is a trivial question; but I just want to make sure: A closed cylindrical container has a capacity of $128\pi \,{\rm m}^3$. Determine the minimum surface area. The answer is $96\pi$. Volume of ...
1
vote
1answer
20 views

Equations of motion - equilibrium condition

Two frictionlessly shiftable mass points are connected by a massless thread of constant length l. For the arrangement given by the figure, use D'Alembert's principle to determine the equations of ...
0
votes
1answer
26 views

D'Alembert's Principle - rocket

Consider a one-dimensional, force-free motion of a rocket with constant mass emission $\mu$ and constant outflow velocity c of the gases. At $t=0$ let $m=m_0$ and $v=0$. a) When is the ...
2
votes
1answer
63 views

Rudin's definition of derivative

Walter Rudin's Principle of Mathematical Analysis defines the derivative as follows in Definition 5.1: Let $f$ be defined (and real-valued) on $[a,b]$. For any $x \in [a,b]$ form the quotient ...
3
votes
4answers
143 views

How to compute the derivative of $\sqrt{x}^{\sqrt{x}}$?

I know have the final answer and know I need to use the natural log but I'm confused about why that is. Could someone walk through it step by step?
2
votes
2answers
127 views

Prove that, $f'(0) \ge -\sqrt{2}$ for a function $f$ satisfying some conditions on $(-1,1)$.

Let $f:(-1,1)\to \mathbb{R}$ be a twice differentiable function such that, $f(0)=1$, $f'(x)≤0$, $f(x)≥0$ and $f''(x)≤f(x)$ for all $x≥0$. Prove that, $f'(0)≥-√2$ Progress: I was able to prove ...
1
vote
2answers
51 views

Is $\max(0, x)$ a differentiable function?

It appears that $\max(x, y)$ isn't differentiable according to this question. However, the explanation is due to the fact that $\max(x, -x) = \lvert x\rvert$, and since there won't be the case ...
0
votes
1answer
33 views

If $t \mapsto \mu((-\infty, t))$ is differentiable at $x$, and $a_n \nearrow x$, does $\mu(\{a_n\})/(x-a_n)$ tend to zero?

Let $\mu$ be a complex (and therefore finite) Borel measure on $\mathbb{R}$ and suppose we have a function $f(t)=\mu((-\infty,t))$. Further suppose that $f$ is differentiable at a point ...
0
votes
0answers
32 views

Differential equation - fractions, circular answer?

Hi this might seem like a really stupid question but then hopefully someone can asnswer it quite easily :) I have function $P{_t}$$=(E{_t}$ $(P{_t}{_+}{_1}+$ $δ{_t}{_+}{_1}$$ )-γΩx$${^*})/$$(1+rf+ψ_t ...
0
votes
2answers
24 views

Comparing the greatest values of two functions (Derivatives)

I've tried doing this task, and for this kind of task I should be using derivatives. When I done all the calculus, everything I got were some weird result which I do not know how to compare. Task ...
0
votes
1answer
36 views

Find antiderivative of $8\sin^3(2x)\cos(2x)$

I was tasked with finding the antiderivative of $8\sin^3(2x)\cos(2x)$ This is what I have $$4\sin^4(2x)-\int24\sin^3(2x)\cos(2x)\,dx$$ I don't know the step after that.
0
votes
1answer
32 views

Finding the tangent line of $x^2 + y^2 = 4$ and $2x^2 + 5y^2 = 10$

If line $y = kx + n$ is a common tangent of circle : $x^2 + y^2 = 4$ and ellipse: $2x^2 + 5y^2 = 10$, then $k^2 + n^2$ is equal to? I've tried doing this task by equalizing derivatives of circle and ...
2
votes
3answers
81 views

Integral of $\sin (x^3)dx$

$$\int \sin (x^3)dx$$ I have tried some substitutions, but I haven't reached the goal... Can you help me?
1
vote
0answers
22 views

differentiability of a function consisting of a bilinear form

Let $A \in M_{n, n}(\mathbb{R})$ be a symmetric matrix, and let $N = \{x \in \mathbb{R}^n \mid x^t A x = 0\}$. I first want to show that $N$ is closed. Next, I want to find an explanation why $f(x) = ...
0
votes
0answers
13 views

Showing that a derivative of a matrix function is surjective

Write each $B$ in $\mathbb{R}^{m\times n}$ in the form $\begin{bmatrix} B_{11} B_{12} \\ B_{21} B_{22} \end{bmatrix}$ such that $B_{11}$ is $k \times k$. Let $U = \{B \in \mathbb{R}^{m \times n} : ...
1
vote
2answers
80 views

Solve $f '(x)=f '''(x) + x$ under initial conditions

I know that: $$\begin{align} f(0)&=4 \\ f '(0)&=0\\ f ''(0)&=3 \\ f '(x)&=f '''(x) + x \end{align} $$ I need to find $f(x)$, how can I solve this? This is obviously a differential ...
0
votes
0answers
14 views

PDF & CDF Differentiation &Integration

Let $F(.)$ be the cumulative distribution function of a $N(0, 1)$ random variable. Define a function $g(x)$ as $$g(x) = \int_{-\ln x}^{x^2}F(t + x) dt.$$ Evaluate $g'(1)$ in terms of $F(.)$, where ...
1
vote
0answers
6 views

Derivative of angular function by cartesian coordinates using Legendre polynomials?

I'm programing some numerical evaluation of force dependent on angle $\phi$ between vector ${\vec a}=(x,y)$ and normalized direction vector ${\hat d}$. To achive maximal performance I wan't to avoid ...
0
votes
1answer
50 views

Norm not differentiable on a dense set of $\mathbb{R}^n$

Would you have an example of a norm on $\mathbb{R}^n$ ($n \ge 2$) which is not differentiable on a dense set?
1
vote
1answer
22 views

Does differentiability of a composite function imply differentiability of all its components?

Does differentiability of a composite function imply differentiability of all its components? I.e. if $f(x)=g(x)+h(x)$ and we know $f(x)$ is differentiable at some point $x=a$, does this also imply ...
0
votes
0answers
24 views

Derivative of function of functions

Is this relationship correct? If so, why? $\frac{\partial h(x)}{\partial x}=\frac{\partial f(g_{1}(x),g_{2}(x)))}{% \partial g_{1}}\frac{\partial g_{1}(x)}{\partial x}+\frac{\partial ...
0
votes
1answer
51 views

What's the derivative of $\ln \lvert \csc x + \cot x \rvert$? [closed]

What is the derivative of $\ln \lvert \csc x + \cot x \rvert$? I've tried to do it and I get really odd numbers, any help showing the steps would be very helpful!
1
vote
2answers
41 views

Clarifications about the correct way to solve exercises (continuity, partial derivatives, differentiability)

I need some clarifications about the correct way to solve an exercise. I have this function: $$f(x,y)=\frac{(x-1)y^2}{\sin^2\sqrt{(x-1)^2+y^2}}$$ and I have to analyse the existence of partial ...
4
votes
2answers
51 views

Intuition of multivariable chain rule

I was learning/reviewing the chain rule for multivariable calculus and was wondering why the multivariable calculus chain rule is a function of summation of products of derivatives rather than just ...
0
votes
2answers
25 views

Why is this matrix function smooth?

Let $A$ be a real, invertible, $k \times k$ matrix, let $B$ be a real $k \times (n - k)$ matrix, and let $C$ be a real $(m-k) \times k$ matrix. How is the function $$ F:(A,B,C) \mapsto CA^{-1}B $$ ...
0
votes
0answers
9 views

Prove that continuity in $x$ of the Gateaux derivative implies Frechet differentiability [duplicate]

Prove that continuity in $x$ of the Gateaux derivative implies Frechet differentiability $f:\mathbb{R}^n\rightarrow\mathbb{R}$ I don't know how to star this problem, I just know the definition ...
0
votes
0answers
10 views

Stationary points of Frobenius Norm

Let $f:\mathbb{R}^{m\times n}\rightarrow\mathbb{R}$ such that $f(X)=\|XB-A\|_F^2$, where $\|\cdot\|_F$ is the Frobenius norm and $A,B$ are constant matrices. I know that $\frac{\partial ...
0
votes
0answers
13 views

Partial derivative of a sum with several subscripts

I am struggling to find the first derivative of a summation with two subscripts. The solution might be simple, but I'm struggling to get the idea. Any hint or help is appreciated. I have the ...
0
votes
1answer
21 views

Gradient of a forth order scalar function with respect to a Matrix

I'm trying to take the gradient of the following function w.r.t A: $$ f(A) = ||AC_YA^T - C_R||_F^2 $$ I tried the following: $$ f(A) = trace((AC_YA^T - C_R)^T(AC_YA^T - C_R)) = ...
4
votes
2answers
91 views

How to compute the derivative of $x^x$ using the definition

I want to prove that $\displaystyle\lim_{h\to 0}\frac{(x+h)^{x+h}-x^x}{h}=x^x(\ln(x)+1).$ If I write $x^x$ as $e^{x\ln(x)}$ I get: $\displaystyle\lim_{h\to0}\frac{e^{(x+h)\ln(x+h)}-e^{x\ln(x)}}{h}$ ...
0
votes
1answer
40 views

Why is $det(Df(x))=0$ for all $x\in U$?

Let $U\subset \mathbb{R}^n$, $f:U\to\mathbb{R}^n$ and $F:\mathbb{R}^n\to\mathbb{R}$ be differentiable functions. Let $\nabla F(y)\not=0$ for all $y\in\mathbb{R}^n$ and $F(f(x))=constant$ for all $x\in ...
0
votes
0answers
13 views

Computing the derivative of a specific matrix function

Let $A$ be a real, symmetric, invertible $k \times k$ matrix and let $B$ be a real $k \times (n-k)$ matrix. How can I compute the derivative of the function $$ F:[A, B] \mapsto B^T A^{-1} B $$ I'm ...
1
vote
2answers
34 views

Find the equation of the parabola with two points and a slope

find the equation of parabola with given two points B (2, 1) and C (4, 3) and slope of the tangent line to the parabola matches the slope of the line goes through A (0, 1.5) and B (2, 1). i have ...
5
votes
1answer
45 views

Is there a nowhere differentiable norm on $\Bbb R^n$?

Is there a nowhere differentiable norm on $\Bbb R^n$? The non differentiable norms that I know (e.g. the one norm $\|\cdot\|_1$ and the infinity norm $\|\cdot\|_{\infty}$) are non differentiable ...
0
votes
0answers
19 views

Differential of $\,f(t,g(t,x))\,$ with $\,g(t,x)=x+c\,t\,$ ($\,c\,$ constant)

I'm looking for the differential of the implicit scalar function $\,f=f(t,x+ct)\,$ where $x$ and $t$ are real variables and $\,c\,$ is a real constant, and this is what I have done: Consider ...
2
votes
1answer
36 views

Derivative Problem: $e^{-x}f(x)$=$2+\int{\sqrt(t^{4}+1)}dt$

Let $f(x)$ be a real valued function defined on $(-1,1)$ such that $e^{-x}f(x)=2+\int_0^x{\sqrt{t^{4}+1}}\;\mathrm{d}t$ for all $x$ belonging to $(-1,1)$ and $g(x)$ be its inverse function. ...
0
votes
0answers
25 views

can any one prove the second branch of the function

can any body tell me how the 2nd branch of the function is defined here $f_t(x)$ is a continuous function \begin{align} f_t(x)= \left\{ \begin{array}{lr} \dfrac{x^t}{t(t-1)\ldots(t-n+1)} & ...
0
votes
1answer
17 views

Bounds on functions via its derivatives

Suppose, we have a function $f$ where $f$ is: Contionuos. Non-Negative Has a derivative given by $f'$. Can we have a bound on $f$ in terms of its derivative $f'$? That is have an inequality that ...
1
vote
1answer
20 views

Finding the Approximate Change of a Point travelling across a parabola.

A point moves along a parabola y^2 = 3x. Find the Approximate change in its distance from the origin as its x coordinate changes from 1 to 1.1 The answer is 0.125 I assume origin is (0,0) ( I am not ...
0
votes
1answer
46 views

$f \colon [a,b] \to [a,b]$ continuous has exactly one fixed point if $f'(x) \neq 1$ for all $x \in (a,b)$

Let $f$ be a continuous function from [a, b] to [a, b], and is differentiable on (a, b). We will say that point y $\in$ [a, b] is a fixed point of f if $y = f(y)$. If the derivative $f'(x) \neq 1$ ...
1
vote
1answer
17 views

Find the critical point using partial derivatives and Hessian matrix

How do I find the critical points, and determine whether they are minima, maxima or saddle point, of the following function: $$f(x,y) = \ln \big( \ (x+y)^2+1 \ \big)$$ For the critical points, I ...
0
votes
0answers
28 views

formula for the derivation

Let $F:\mathbb{R}^2\to\mathbb{R}$, $g:I\to\mathbb{R}$ be differentiable functions, $I\subset \mathbb{R}$ open interval. It is $F(x,g(x))= 0$ and $D_2F\not=0$. I have to deduce a formula for the ...
0
votes
1answer
16 views

A fundamental theorem of calculus problem

I'm asked to prove that $N_x - M_y = f_3(x, y, z)$, but: $N(x, y, z) = \int_{x_0}^{x}f_3(u, y, z_0)du - \int_{z_0}^{z}f_1(x, y, u)du$ $M(x, y, z) = \int_{z_0}^{z}f_2(x, y, u)du$ So: ...
0
votes
1answer
40 views

Integral using trig substitution

The problem says to find the arc length of the curve which isn't that bad. I know the formula for the arc length of $r(t)$ is: $\int_a^b |r'(t)|\,dt$. The equation is this: $x=\cos(3t), y=\sin(3t), ...
10
votes
3answers
121 views

Let $f$ be a continuous and differentiable function such that $f(a)=f(b)=0$ , show that $f'(c)=\pi f(c)$ for some $c \in (a,b)$

Let $f:[a,b] \rightarrow \mathbb{R} $ be a continuous function in $[a,b]$ and differentiable in $(a,b)$ such that $f(a)=f(b)=0$ . Show that $f'(c)=\pi f(c)$ for some $c \in (a,b)$ The initial ...
4
votes
1answer
57 views

What is so good about the $L^2$-norm of the second derivative being small?

One of the main properties of cubic splines is the minimality property which basically means that if $s$ (cubic spline) and $g$ (some other function) interpolate $f$ in a certain way then $$\Vert s'' ...
1
vote
1answer
17 views

Use the chain rule to find the derivative of a multivariable function?

I know that $\frac{dg}{dt} = (2xy , x^2)$ Is $\frac{d}{dt} g(r(t))$ simply equal to $\frac{dg}{dt}$ evaluated at $r(t)$? If so, how would I calculate this? $g(x,y)$ depends only on $x$ and $y$, ...