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

learn more… | top users | synonyms (2)

0
votes
1answer
87 views

Derivative of a trace with respect to a matrix when the matrix is implicitly defined

I am trying to solve the following matrix maximization problem $\max_\Theta trace (A H (\Theta, P))-ln(det(H (\Theta, P)))-ln(det(P))$ , where $A, \Theta, P, F$ are all matrices and $P$ is ...
0
votes
1answer
25 views

Particular maximal solution of the inhomogeneous equation $y^{'} = |t|y + t$ on $\mathcal I = \mathbb R$.

Particular maximal solution of the inhomogeneous equation $y^{'} = |t|y + t$ on $\mathcal I = \mathbb R$. I've already solved this system in the homogeneous case $b(t)=0$, where I've found the ...
1
vote
1answer
28 views

Is there a difference between “in the direction of (1.1)” and “in the direction toward (1.1)?”

The question that I have is A differentiable scalar field f has, at the point (1.2), directional derivatives 2 in the direction toward (2.2) and -2 in the direction toward (1.1) Determine the ...
2
votes
0answers
16 views

Find points with slope parallel to function

Find all points (a, b) on the curve y = x^3 - x + 1 where the tangent line is parallel to the line y = 11x + 5. Just need confirmation on how to complete this problem. You take the slope of the ...
1
vote
0answers
25 views

First order partial derivatives and ball centered at $a$ [duplicate]

Suppose $r>0$, $a\in\mathbb{R}^n$, and $f\colon B_r(a)\to\mathbb{R}^m$. If all first order partial derivatives of $f$ exist on $B_r (a)$, and $f_{x_j}(x) = 0$ for all $x\in B_r (a)$ and all ...
1
vote
0answers
20 views

Find the line tangent to the parametric curve $\left\langle t^3-1,t^4+1,t \right\rangle$

Firstly, this is a homework problem, so I would appreciate it if you might not just write the answer and rather, if I am wrong, provide suggestions only. I am given a parametric curve with the ...
4
votes
1answer
277 views

Is there a function that is differentiable but not integrable? [closed]

is there such a function that is defined in [0,1], differentiable in (0,1) but not integrable? Thanks in advance.
1
vote
1answer
42 views

function not continuous, partial derivatives exists -> partial derivatives not continuous

I'm a bit confused about this. I know that if all partial derivatives exist it's not necessary for function to be differentiable. Usual examples for non differentiable function for which all partial ...
2
votes
0answers
37 views

Is every Volterra's function unbounded?

Volterra's function is a function $f\colon\mathbb{R}\to\mathbb{R}$ such that: $V$ is differentiable, $V'$ is bounded, $V'$ is not Riemann-integrable. ...
2
votes
2answers
54 views

Similar problem to Taylor's theorem proof

43. Let $a_1,\dots,a_{n+1}$ be arbitrary points in $[a,b]$ and let $$Q(x)=\prod_{i=1}^{n+1}(x-x_i).$$ Suppose that $f$ is $(n+1)$-times differentiable and that $P$ is a polynomial function of ...
1
vote
0answers
27 views

derivative vanishing on a dense set

$f\colon(0,1)\to\mathbb{R}$ is strictly increasing and differentiable. is it possible that $\{x\colon f'(x)=0\}$ is dense in $(0,1)$ ? I found the construction of a singular function. But singular ...
0
votes
2answers
100 views

How to find maximum and minimum value of a variable in 2 variable equation

I have been given a equation $$4x^2 + 12xy + 10y^2 -4y +3= 0$$ How can I find maximum value of $y$ is this intermixed variable equation ? I have not been Introduced to multi variable calculus . can ...
2
votes
0answers
38 views

Bochner integral vs regulated integral

I'm reading Serge Lang's Real And Functional Analysis and at some point he introduces the regulated integral in order to prove the Fundamental Theorem Of Calculus (in the context of Banach Spaces), or ...
0
votes
2answers
39 views

differentiability at a point (0,0) based on partial derivatives

For $$ f(x,y)=\begin{cases} y^2 sin\left(\frac{x}{y}\right) & \text{if } y\neq0 \\ 0 & \text{if } y=0 \end{cases}$$ i've shown that it is continuous and that the partial derivatives ...
0
votes
0answers
139 views

Differentiation of an integral with lim

I'm looking at a proof that wants to show for $d\geq3$, $y\in\mathbb{R}^d$ and $x\in B_r(0)$ $$\lim_{h\rightarrow 0} \int_{B_r(x)} \frac{y_j}{h}\left(\frac{1}{|y-hu_i|^d}-\frac{1}{|y|^d}\right) ...
3
votes
2answers
367 views

Find $f'(8.23)$ where $f(x)=23|x|−37\lfloor x\rfloor+58\{x\}+88\arccos(\sin x)−40\max(x,0)$

Let $$f(x)=23|x|−37\lfloor x\rfloor+58\{x\}+88\arccos(\sin x)−40\max(x,0).$$ Find $f^\prime(8.23)$. Note: For a real number $x$, $\{x\}=x−\lfloor x\rfloor$ denotes the fractional part of x. I don't ...
0
votes
1answer
26 views

How to evaluate the derivative of a function defined by different formulas in different intervals?

Let $f$ be a function defined as following: $f(x) = e^{\frac{-1}{x^2}}$ if $x \neq 0 ; f(0) = 0$ How should I proceed to evaluate it's derivative in $x= 0?$
2
votes
1answer
94 views

Can the chain rule be proven by math induction?

I need to prove the chain rule for a math project and I am wondering if it can be proven by math induction. If not, how can this rule be proven?
0
votes
2answers
45 views

Finding Derivative $f(x)=|x-3|$

I need to find the derivative from the right and the left like, $$f'_-(x) = \lim_{\Delta x \to 0^-} {f(x_1 + \Delta x) - f(x_1) \over \Delta x} \\ f'_+(x) = \lim_{\Delta x \to 0^+} {f(x_1 + \Delta x) ...
0
votes
1answer
23 views

Operator on polynomials, antiderivative

We are given a linear map: $$\mathbb{R} [X] \ni p \rightarrow q \in \mathbb{R} [X], \ \ q'=p, \ \ q(0)=0 $$ and two norms on $ \mathbb{R} [X]$ : $||p||_{\infty} = \sup _{t \in [0,1]} |p(t)|$, ...
0
votes
3answers
77 views

Derivative of the function $f(x) = |x+2|$

How can I get the equation of the derivative $f(x) = |x+2|$ ? I have already graphed the original function $|x+2|$ and the derivative function, but I'm not sure how to find the derivative, the ...
0
votes
3answers
43 views

Need help with inverse trig functions

How can I prove $\arccos\frac{1-x^2}{1+x^2}$ is equal to $2\arctan x$ for $x\geq0$ ? I am also supposed to use the fact that if a function is defined and differentiable in $(a,b)$ and $f '(x) = 0 ...
1
vote
0answers
23 views

Term for a Convex Function whose derivative is also convex

Let $f(x)$ be a monotone non-decreasing convex function such that its derivative $\frac{d}{dx}f(x) = f'(x)$ is also a convex function. Is there a term in literature that is used to refer to such ...
-1
votes
1answer
17 views

Continuous composed with differentiable

If $f(x)$ is $C^\infty$ and $g(x)$ is bounded and continuous does that imply that $f(g(x))$ is differentiable
3
votes
3answers
139 views

Implicit differentiation of $e^{x^2+y^2} = xy$

I just want to reconfirm the steps needed to answer this question. Thank you Find $\dfrac{dy}{dx}$ in the followng: $$e^{\large x^2 + y^2}= xy$$ I got this so far. ...
2
votes
1answer
44 views

derivative of sqrt(5/(x+7))

Why is it that: $$\frac{d}{dx}\sqrt{\frac{5}{x+7}} = -\frac{\sqrt{5}}2\frac{1}{(x+7)^{3/2}}$$ (image) ??? My attempt: It seems that somehow you end up adding 1 to 1/2 to get 3/2 in the exponent. ...
1
vote
1answer
41 views

Finding the tangent line through the origin

Find the tangent line to: $$f(x) = \sqrt{x-1}$$ that passes through the origin $(0, 0)$. $$f'(x) = \frac{1}{2\sqrt{x-1}}$$ The line will be tangent at $(a, b)$ so then: $$f'(a) = ...
-2
votes
2answers
57 views

Integral $\int x^7\cos x^4 dx$

$\displaystyle \int x^7\cos x^4 dx$ I tried first by letting $x^4 = u$ and then using integration by parts by assigning f(x) to $u^\frac74$ and cos(u) to g'(x) and I end up getting after applying ...
3
votes
4answers
59 views

$dx$ being a desginator (with respect to $x$) or being a term?

I am confused as to what $dx$ truly is. I am doing some u-substitution problems and this is what I came across: $$\int 2x(x-1)^{1/2}\,dx$$ $u=x-1$ and therefore $du=1$ when we substitute we get: ...
2
votes
2answers
52 views

derivative of $\sec^2(x/12)$

Alright, so the derivative of $\sec^2(x/12)$ is $\frac{1}{6} \tan\left(\frac{x}{12}\right) \sec^2\left(\frac{x}{12}\right)$ But if you use chain rule, you get: $$2 \sec\left(\frac{x}{12}\right) ...
2
votes
1answer
94 views

Example of non-differentiable continuous function with all partial derivatives well defined

Give an example of a function $f : \mathbb{R}^3 \to \mathbb{R}$ such that the partial derivatives exist at $(0,0,0)$, and $f$ is continuous at $(0,0,0)$, but it is not differentiable at $(0,0,0)$. Any ...
0
votes
2answers
32 views

differentiability and continuity in R3

Prove that if a function is differentiable at $(a,b,c)$ in $\mathbb R^3$ then it is continuous at $(a,b,c)$. I tried to imitate the proof that if $f$ is differentiable at a specific point in $\mathbb ...
1
vote
0answers
19 views

Calculate Laplace transform of the product of t and f(t) by differentiating f(t) (5.5-8)

Request: Please check my work. State where errors, if any, occurred and how to correct them. Is there a better way to calculate the transform other than the present method given? Given: Find the ...
0
votes
2answers
46 views

Prove that $f$ has derivatives of all orders at $x=0$ [duplicate]

Let $\displaystyle f(x) = \begin{cases}e^{- \frac{1}{x^2}} &\text{for } x \neq 0 \\ 0 & \text{when } x=0 \end{cases}.$ Prove that $f$ has derivatives of all orders at $x=0$, and ...
1
vote
1answer
17 views

Calculate Laplace transform of the product of t and f(t) by differentiating f(t) (5.5-6)

Request: Please check my work. State where errors, if any, occurred and how to correct them. Is there a better way to calculate the transform other than the present method given? Given: Find the ...
2
votes
1answer
19 views

Calculate Laplace transform of the product of t and f(t) by differenitating f(t) (5.5-4)

Request: Please check my work. State where errors, if any, occurred and how to correct them. Is there a better way to calculate the transform other than the present method given? Given: Find the ...
0
votes
1answer
34 views

Differentiability in $\mathbb R^3$

$G$ is an open subset of $\mathbb R^3$ and $(a,b,c)$ belongs to $G$. $f$ is a function from $G$ to $R$. i) Define: $f$ is differentiable at $(a,b,c)$ ii) Prove if $f$ is differentiable at $(a,b,c)$ ...
0
votes
4answers
77 views

Derivative of $f(x) = x^5$ using the definition.

Let $f(x)=x^5,$ and $\quad P(1,1)$ $(a = 1,\text{ and } f(a) = 1)$. $$\lim_{h\to 0} \frac{f(a+h) - f(a)}h \implies\lim_{h\to 0} \frac{(1+h)^5 - 1}h=\lim_{h\to 0} \frac{1}h((1+h)^5-1)$$ After This ...
1
vote
3answers
144 views

Multivariable Calculus, rate of change.

An insect is moving on the ellipse $2x^2+y^2=3$ on the $xy$-plane in the clockwise direction at a constant speed of 3 centimeter per second. The temperature function $T(x,y)$ (experienced by the ...
1
vote
0answers
13 views

Linearlized curvature operator

While reading a paper, I came across the term for linearized curvature operator \begin{eqnarray} \kappa_1 = -\frac{1}{{(1+x^2)^{\frac{3}{2}}}}\frac{\,d }{\,d x^2} + ...
0
votes
1answer
83 views

How to find the minimum value of this integral?

I am struggling to find the solution to this problem. If anyone could help to explain how to solve this problem to me, it would be really appreciated. Let $$ f(x)=-\sqrt{3}x+(1+\sqrt{3}) $$ $$ ...
0
votes
3answers
93 views

Find 6th derivative of $(\cos(5x^2)-1)/x^2$ at $x=0$

Let $$ f(x)=\frac{\cos(5x^2)-1}{x^2} $$ We want to compute the $6th$ derivate of $f(x)$ at $x=0$. Using a calculator, I found $18750$ (which is correct). But I don't understand how to find this ...
0
votes
1answer
31 views

Lyapunov and Asymptotically stability

How do you determine if a function is Lyapunov or asymptotically stable? The definitions do not seem to tell us how to prove whether a solution is stable or unstable. For example, I am trying to ...
0
votes
1answer
62 views

How to formalize that $\lim\limits_{x \to +\infty} \frac{f(x)}{g(x)} = 0 \implies$ $g$ “grows faster” than $f$?

I understand that $\lim\limits_{x \to +\infty} \frac{f(x)}{g(x)} = 0$ implies that, for sufficiently large values of $x$, $f(x)<g(x)$, as a direct consequence of the definition of limit to ...
1
vote
1answer
48 views

Solve $h(x)+h'(x)(8-x)-32=0$ for x.

Solve $h(x)+h'(x)(8-x)-32=0$ for $x$.Where $$h(x)=\frac{\frac{1}{16}x^2 - 2 x + 80}{\left(\frac{1}{16}x^2 - 2 x + 20\right)^2}$$ Should I go with characteristic equations? or is there another way. ...
0
votes
1answer
37 views

What is the $n$th derivative of $\coth(x)$?

I would like to know the $n$th derivative of the Hyperbolic Cotangent, i. e., $\frac{\partial^n}{\partial x^n} \coth( x )$. So far, I have only found an expression for the $n$th derivative of the ...
0
votes
1answer
71 views

Give the differential and the derivative of $f(X) = I − X(X^tX)^{-1}X^t$

I don't know what to do, maybe use the product rule. Give the differential and the derivative of the function $$f(X) = I − X(X^tX)^{-1}X^t $$
0
votes
1answer
50 views

Question on a special Derivative

I have this functional defined from a Hilbert space $H$, $J\colon H\rightarrow \mathbb{R}$ defined by: $$ J(u)=\frac12 \|u\|^2-\int_0^1(A(su),u) ds $$ where $A\colon H\rightarrow H$ is a potential ...
2
votes
3answers
57 views

General solution to ODE $ y''-Ay^5=0 $

What is the solution of $$ y''-Ay^5=0 $$ I got the solution $ y = {(3/4A)}^{1/4} x^{-1/2}$ using trial and error but how to solve this type of problem in general?
0
votes
1answer
31 views

An ant is walking up a hill. at what x does he see the blade of grass.

've been working on this problem with Mathematica and by hand-help with either would be fantastic. The blade of grass is given by the line segment from (32,1/5) and (32,8). The 2D hill is given by ...