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

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2
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2answers
55 views

Properties inherited by $f\circ g$ from $f$

Suppose $f,g:\mathbb{R}\to \mathbb{R}$ Prop: Suppose $g$ and $f \circ g$ are ______, and $g$ achieves every value in $\mathbb{R}$. Then $f$ is ______. If in the blanks we put the word ...
0
votes
1answer
12 views

Sufficient criterion for a function in C to be differentiable

Give a sufficient criterion for a function f(z), z $\epsilon$ C to be differentiable at $z=z_0$. I know that continuity does not imply differentiability, can't think of a criterion that implies ...
0
votes
3answers
48 views

Implicit differentiation of trig functions

I'm struggling somewhat to understand how to use implicit differentiation to solve the following equation: $$\cos\cos(x^3y^2) - x \cot y = -2y$$ I figured that the calculation requires the chain ...
1
vote
1answer
24 views

Solve given qus without using partial fraction method

$$z=f\left(x,y\right)=x^{2}\tan^{-1}\left(\frac{y}{x}\right)-y^{2}\tan^{-1}\left(\frac{x}{y}\right)$$ Prove that $$\frac{\partial^{2}f\left(x,y\right)}{\partial x\,\partial ...
0
votes
4answers
158 views

derivative of $x\cdot|\sin x|$

I have the function $f(x)=x|\sin x|$, and I need to see in which points the function has derivatives. I tried to solve it by using the definition of limit but it's complicated. It's pretty obvious ...
6
votes
3answers
78 views

Interesting, unusual max/min problems?

So I've got to that stage of my elementary mathematics subject for engineers when we talk about differentiation and solution of max/min problems. And I'd like to entertain and engage the students ...
1
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2answers
57 views

Successive Differentiation of $\mathrm{e}^{g(t)}$

I am trying to find the closed for solution for $A_n$. Assume $A_0 = g'(t)$, $A_1 = g'(t)$, and $$\dfrac{d^n}{dt^n}\left[e^{g(t)}\right] = A_n e^{g(t)}$$ The problem has a recursive relationship of ...
0
votes
1answer
22 views

Calculate derivative of multicase function involving exponentials as $x \to 0$ by definition

While this seemed (and probably does seem to some of you) like a simple question a first it stumbled me a bit. We were asked to calculate the derivative of: $$f(x) = \left\{ \begin{array}{lr} ...
1
vote
1answer
32 views

Prove there's $M>0$ such that: $f(x)\le Mx^2$

Let $f:[-1,1]\to\mathbb{R}$, three-times differentiable function and $f(0)=0$, $f(x)\ge 0$ for all $x\in[-1,1]$. Prove there's $M>0$ such that $f(x)\le Mx^2$. Hint: use Taylor formula. So ...
2
votes
0answers
87 views

A Tricky Weak Derivatives question

I recently came across the following statement and am having trouble proving it correct. I wonder if you could help. Given a weak derivative, $x'$, there exists an absolutely continuous ...
0
votes
0answers
28 views

Derivative with respect to a function

We have a function ${f(s,{\psi(s)}_{3\times 1})}_{3\times1}\tag1$ Given Data $f,\psi$ are matrices and their dimensions are already given in the question s is not a matrix, it is a scalar ...
1
vote
1answer
38 views

Differentiation under the integral sign (one complex variable)

Let $u(z), u'(z)$ be complex-analytic functions on an open neighborhood $\Omega \subseteq \mathbb{C}$ of the origin. Also, let $f(X)$ be a complex-analytic function. For $s \in [0,1],$ define $$g(s,z) ...
0
votes
0answers
46 views

Function with constant derivative

We have a column matrix $P_i$ defined as follows $P_i= {\begin{pmatrix} a_i \\ b_i \\ c_i \end{pmatrix}}_{3\times 1}\tag 1 $. Given Data All $a_i,b_i,c_i$ are constants It is given that $i$ can ...
1
vote
1answer
54 views

Finding the derivative of $y=12x^4\sqrt[3]{x^2}-2e^x+9$

Let $$ y=12x^4\sqrt[3]{x^2}-2e^x+9 $$ How can we find $y^\prime$?
0
votes
1answer
46 views

How to derive “Pooled Sample Variance”?

Let $s_p^2 = bs_1^2 + (1-b)s_2^2$, this can be an unbiased estimator of population variance, provided we find the correct value for $b$; in particular, $s_p^2 = \frac{(n1-1)s_1^2 + ...
0
votes
1answer
41 views

Derivative of an integral with variable in upper bound and a term of the integrand

So I want to take the first and second derivatives of a function g(Z) which is made up of several terms, one of which is where Z and H are our variables. Taking the derivative of this, it seems ...
1
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0answers
50 views

Derivatives of Lagrange polynomials

It seems there is some relationship between Lagrange polynomial and Legendre polynomial. That is Lagrange polynomial can be expressed as a function of Legendre polynomial. If so, I could use this ...
0
votes
2answers
46 views

Calculus - Derivative help.

I need to get dervative for this function $$\sqrt{1+\sqrt{x}}'$$ I used $(f+g)'(x) = f'(x) + g'(x)$ so: $$\sqrt{1} + \sqrt {\sqrt{x}}$$ So : $$\sqrt{1}' = 1' = 0$$ $$\sqrt {\sqrt{x}'} = \sqrt ...
-2
votes
2answers
105 views

How do i construct $C^\infty$?

I'm trying to define $C^\infty$ rigorously and i have a trouble with this. Mathematical Induction should be used, but i dunno where to apply this. I'm going to illustrate what i tried below: Before I ...
0
votes
1answer
52 views

If function $f$ has zero value and positive derivative at both endpoints, then $f''(\eta)=f(\eta)$ for some $\eta$ [duplicate]

Suppose $f(x)$ is differentiable on $[a,b]$, twice differentiable in $(a,b)$. Given that $f(a) = f(b) = 0$, $f'(a)f'(b) > 0$, Prove that $\exists \zeta \in (a,b), f(\zeta) = 0$ and $\exists \eta ...
0
votes
1answer
34 views

Differentiate $(x-1)^2 \sin x$ where $x$ is in radians

How would I differentiate, simplify and then find $f'(\pi/2)$: $$ f(x)=(x-1)^2 \sin x $$ I'm not sure how to differentiate $\sin x$ to then use it later to find an answer, any help would be much ...
1
vote
2answers
53 views

Second derivative using limits

If f is a function that is two times differentiable at x = a then: $\lim\limits_{h \to 0} \frac{f(a+h)-f(a)-hf'(a)}{h^2/2}=f''(a)$ I don't know how to prove or disprove this. I know I have to use ...
0
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1answer
25 views

Non-differentiability of $\max \limits_i f(i)$

How can we formally show that $\max$ and $\min$ functions are non-differentiable? In particular, I was looking at the L1 matrix norm defined as: $\|A\|_1 = \max \limits_{i \le j \le n} \sum ...
3
votes
3answers
75 views

Find the limit and derivative of integral function.

$\psi_m(x)$ is defined as $$\int_0^{\ln|x|}e^{mt}\sin(t)^m\mathop{dt}$$ with $m$ a natural number greater then zero. Now the question is, does $\lim\limits_{x\to 0}\psi_m(x)$ exist. I've tried using ...
5
votes
1answer
83 views

Derivative of $\frac{x}{f(x)}\frac{df}{dx}$

Suppose we have a function $f(x):\mathbb R^+\to\mathbb R^+$ that satisfies: 1) $0\leq\frac{df}{dx} \leq 1$ 2) $f(0) = 0$, then do we have $$\frac{d}{dx}\left(\frac{x}{f(x)}\frac{df}{dx} ...
0
votes
1answer
78 views

Using bordered Hessian matrix to determine non-degeneracy and type of constrained extremum

I have the following problem: $\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}\def\g{g(x_1,x_2,x_3)}\def\l{\lambda}\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}$ Find the ...
0
votes
1answer
21 views

How do I find the ridges and valleys given a surface elevation function

Given a surface with a single elevation value for every x and y how can I find the places where the isoelevation contours have the tightest bends? And how can I differentiate between bends that are ...
0
votes
1answer
44 views

A rigorous proof of continuous differentiability

This small step comes from my reading on saddle point approximation: suppose $$ w=\text{sign}(s)\sqrt{2(s K'(s)-K(s))}\tag{*} $$ where $K$ is convex with and continuously differentiable for all orders ...
2
votes
1answer
61 views

What is an intuitive way to see $\frac{d}{dx}\sin^{-1}x+\frac{d}{dx}\cos^{-1}x=0$?

Without calculation, explain why $\frac{d}{dx}\sin^{-1}x+\frac{d}{dx}\cos^{-1}x=0$?
4
votes
3answers
66 views

Differentiate with product rule

Question: differentiate $x(x^2 +3x)^3$ I have gotten to the point where i've used the product rule and i've gotten $$(x^2 + 3x)^3 + x\cdot(3x+9)(x^2 + 3x)^2$$ but now that it comes to the ...
2
votes
2answers
61 views

Find the Derivative

I'm currently studying the product rule and have come across a section of questions that seems to make no sense. I'm sure there's just one little thing that I'm missing but I am unable to spot it. ...
0
votes
2answers
113 views

Definite integral-dot product

I have an integral equation containing dot product $$\int_{0}^{L} \left(\frac{a}{L}.b(s)\right)\mathrm ds\tag 1$$ Data Given a is a constant vector of size 3 b(s) is a varying vector of size 3 " . ...
1
vote
4answers
55 views

Proving the Derivative of $f'(x) = b^x$

Given $f(x) = b^x = e^{x\ln b}$ for $b > 0$, can someone show me how $f'(x) = \ln b e^{x\ln b}$ ?
0
votes
0answers
25 views

Sufficient conditions for the objective function to have gradient pointing towards the origin

Say I have a sufficiently smooth objective function $J(x)$. How can I ensure the below statement is fulfilled -with additional assumptions to $J$ if needed. $\underset{{{| x ...
1
vote
2answers
101 views

How to obtain probability distribution from the generating function $G(s) = e^{a(s-1)^2}$?

I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this: $G(s) = e^{a(s-1)^2}=\sum s^np(n)$ I need first to do Maclaurin expansion of the exponential and ...
3
votes
1answer
75 views

Computing the derivative of a transformation matrix

I am trying to find a geometric transformation between two images, where the transformation is a simple scaling matrix. So, if I denote the two image functions as $r$ and $f$ and the scaling matrix as ...
7
votes
5answers
210 views

Is there an easier way to find $\frac{\mathrm d^9}{\mathrm dx^9}(x^8\ln x)$ than using the product rule repeatedly?

Find $\dfrac{\mathrm d^9}{\mathrm dx^9}(x^8\ln x)$. I know how to solve this problem by repeatedly using the product rule, but I was wondering if there is a short cut. Thanks.
0
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0answers
30 views

Applications of Continuity and Differentiability on a Tough Qn

Given f is cont on [0,1] and that it is twice differentiable on (0,1). Suppose that Integral from 0 to 1 of f(x) dx = f(0) = f(1). Prove that there exist a number c where c is an element of (0,1) ...
0
votes
3answers
38 views

For what values of $r$, $x^r$ has infinite slope at $x=0$?

I'm learning calculus form MIT OCW 18.01SC. In session 23 (it's about linear approximation), prof computes linear approximation near $0$ of some basic functions. $$\sin{x}, \cos{x}, e^x, \ln{(1+x)}, ...
0
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1answer
24 views

Baby version of Sturm Comparison Theorem

In Problem 15-32 of Spivak's Calculus, 4th edition, he proves the following: Suppose $\phi_1$ and $\phi_2$ satisfy $$\phi_1''+g_1\phi_1=0, \\ \phi_2''+g_2\phi_2 = 0,\\[10pt] g_2>g_1, \\[10pt] ...
0
votes
2answers
39 views

Finding the partial derivatives of $V (x, y) = U (x, y)e^{−ax−by}$

I think I did something wrong, so I was hoping someone might be able to show me the solution Two functions $V (x, y)$ and $U (x, y)$ are connected by the equation $$V (x, y) = U (x, y)e^{−ax−by}$$ ...
1
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1answer
53 views

Two methods of solving the differential equation $y' = .75 -.005y$

I am working on a differential equation problem and I am stumped since two different methods seem to give me two different answers Method 1 Given $\frac{dy}{dx} = .75 -.005y$ ...
0
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1answer
46 views

How would I use derivatives for suggesting an option to my user?

I was learning derivatives. I understood the theoretical concept behind it. When I was searching for the real-life example in machine learning I came across one of the answers in this question How do ...
0
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2answers
55 views

Find equation of tangent line to a curve $g(x)$ at $x=4$

So I am trying to find the equation of a tangent line to the curve: $$y = g(x)\text{ at }\,x = 4$$ given $g(4) = -6,\;$ and $\;g'(4) = 2$.
0
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2answers
160 views

Least Squares: Derivation of Normal Equations with Chain Rule

I'm new to Stackexchange so please bear with me. I'm struggling with the least squares formula. Now Wikipedia does show ways to derive the "normal equations". But I'd like to get the same result ...
0
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2answers
36 views

Distributional derivate of $f(t)$

I have the function $f(t)=e^{-|t|}$ And I want to distribution derivate it to $f''(t)$. I am aware of that the $f'(t)$ function will be: But how do I derivate to $f''(t)$ ?
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0answers
54 views

How can I calculate this matrix differentiation?

I am studying about the Matrix Differentiation. I don't know if this red box differential metric, which is how it is calculated.
3
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2answers
117 views

Does my proof of $|x+y| \le |x| + |y|$ make sense? How do I conclude a proof?

Thank you for reading it. I know I made a lot of mistakes. This is my first ever proof that I have attempted. Another note is that I only have been studying proofs for about a week. Any advice will be ...
2
votes
0answers
57 views

$n$th derivative of $f(x)$ using limit definition

After playing around with the limit definition of the derivative for higher order derivatives, I noticed the following odd relationship to determine it for an nth order derivative: Let $F^n=f(x+nh)$ ...
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0answers
20 views

Prove that the maximums of the family$f(x)=xe^{-ax},\ a>0$ are collinear.

If $f(x)=xe^{-ax}$, and a is an integer constant greater than zero, then $$\frac{df}{dx}=(1-ax)(e^{-ax}).$$ The maximum of $f(x)$ would then be at the $x$-value where $\frac{df}{dx}=0$. Since ...