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

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1
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0answers
78 views

Problem with trigonometric substitution proof

I'm sad, I can't get it. I know perfectly how to integrate using the mechanical process described in the books, but I want to understand the proof of it. My book (Stewart) says: In general we can ...
2
votes
2answers
24 views

How to find local maximum of the function $f(x) = x^3-9x^2+24x+4$?

Give the value of x where the function $f(x) = x^3-9x^2+24x+4$ has a local maximum. a) -4 b) 4 c) 2 d) 3 e) -2 I graphed it and I'm not sure how to find the local max
0
votes
0answers
26 views

Partial derivatives of a piecewise defined function

If a function f as $$f(x,y) =\begin{cases} x^2+2x+5y+10 & \text{ for } (x,y)\neq (0,0) \\ y^2+2y+x+10 & \text{ for } (x,y)=(0,0) \end{cases}$$ Is it true that $$f^\prime_x(x,y) ...
3
votes
1answer
31 views

Studying the function $f(x) = x^4-6x^2$ using derivatives: minima, maxima, inflection, concavity

(I know this is my second question today, but I'm explaining what I'm doing so I hope it's okay) Consider the graph of $f(x) = x^4-6x^2$. a) Find the relative maxima and minima (both x and y ...
1
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5answers
54 views

Derivative of $\ln(1+\sin 2x)$

Differentiate the equation $y=\ln(1+\sin2x)$. It will be something to do with the $\frac{d}{dx}\{\ln\:x\}=\frac{1}{x}$ rule, but I'm not sure how to deal with the $\sin2x$ term.
0
votes
1answer
41 views

derivability don't imply partial to be continuous ? example

Is $$f(x,y) =\begin{cases} x^2+2x+2y & \text{ for } (x,y)\neq (0,0) \\ y^2 & \text{ for } (x,y)=(0,0) \end{cases}$$ derivable? But its partials are not continuous?
1
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0answers
21 views

How t find z (unknown) in Runge-Kutta question

I'm trying to solve the below question solve $\dfrac{dx}{dy}=\dfrac{1}{x+y}$ for $x=0.5$ to $z$ using R-K (order $4$) with $x_0=0$, $y_0=1$ (take $h=0.5$). Please help me and tell me how to ...
1
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1answer
52 views

Derive the formula for the sum of the first $n$ squares using derivatives and integrals

I wanted to prove the formula for sum of squares without using induction and thought using derivatives would be the easiest approach ...
0
votes
1answer
30 views

continuity, discontinuity derivative and relation to being derivative but its partials are not continuous

is $$f(x) =\begin{cases} x^2\sin(\frac{1}{x}) \mbox{ for } x\neq 0 \\ 0 \mbox{ for } x= 0\end{cases}$$ a continuous function specially at point x=0? And why being derivable its derivative is not ...
0
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2answers
32 views

Differential identity and wedge products

Apparently $dx^{i_1} \wedge ... \wedge dx^{i_k}=d(x^{i_1}dx^{i_2}\wedge ... \wedge dx^{i_k})$ which I cannot see proved anywhere in my notes. It just stated as if it is obvious which I don't believe ...
2
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0answers
20 views

Derivative of the Inverse Cumulative Distribution Function for the Standard Normal Distribution

As the title says, I am trying to find the derivative of the inverse cumulative distribution function for the standard normal distribution. I have this figured out for one particular case, but there ...
0
votes
1answer
17 views

Minimizing an error function by deriving a system of linear equations

Consider the following formula: $$E(\mathbf{w}) = \frac{1}{2}\sum_{n=1}^{N}\{y(x_n,\mathbf{w})-t_n\}^2$$ where $\mathbf{w}$ is a vector of weights; $x_n$ and $t_n$ come from two vectors of length ...
2
votes
2answers
62 views

Part of proof that $d^2\omega=0$

The following comes from the proof in differentiable manifolds that $d^2\omega=0$. Let $f$ belong to the set of $0$-forms. From definition I have that $\displaystyle df = \frac{\partial f}{\partial ...
1
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1answer
27 views

Show that the value of $\frac{\text{d}^{2r+1}y}{\text{d}x^{2r+1}}$ when $x=0$ is $\frac{1}{2^{2r}}\left(\frac{(2r)!}{r!}\right)^2$

The question originally asks you to prove that if $y=\sin^{-1}(x)+(\sin^{-1}(x))^2$ that: $(1-x^2)y''-x y'$ is independent of $x$. I get that $(1-x^2)y''-x y'=2$ hence proving the first part. The ...
0
votes
0answers
15 views

Which property can be used to derive a differential equation for a reparametrization

With $0\le t\le1$, two space curves given by: $$c_1(t)=(1,t,0)\quad\quad c_2(t)=(0,t,2t(1-t))$$ One of them, say $c_1$, must be reparametrized by $r(t)$ in order to minimize the area between the ...
1
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1answer
32 views

Differentiability of non-analytic complex functions

Any complex function that is analytic on an open set is differentiable on that set. But can a function fail to be analytic on an open set but still be differentiable? For example, the function ...
2
votes
2answers
51 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
7 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
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3answers
44 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 ...
0
votes
1answer
23 views

Limit involving directional derivatives [on hold]

If $z = f(x, y)$ is differentiable at $\textbf{x}_0 = (x_0, y_0)$ Find $ \lim\limits_{\textbf{x} \to \textbf{x}_0}\dfrac{f(\textbf{x})-f(\textbf{x}_0)-\nabla f\left(\textbf{x}_0 \right)\cdot ...
1
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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
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4answers
144 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
59 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
49 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
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1answer
20 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
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1answer
30 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
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0answers
57 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
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0answers
25 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
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1answer
18 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
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0answers
39 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
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1answer
50 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
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1answer
25 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
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1answer
18 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
18 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
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2answers
38 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
98 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
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1answer
51 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
33 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
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2answers
50 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
22 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
73 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
81 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
32 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
16 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
38 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
60 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
60 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
59 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
51 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
45 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 $f'(x) = \ln b e^{x\ln b}$ ?