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

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4
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3answers
56 views

Problem in understanding the concept of differentiation.

I have just completed the chapter differentiation. But I still have confusion in understanding the concept of it.I observe a fact that if we can draw a tangent to a curve at any point of it or in ...
1
vote
1answer
51 views

Derivative of $\operatorname{tr}[(CC^{T})^{-1}]$?

What is the derivative of $$\operatorname{tr}[(CC^{T})^{-1}]$$ with respect to the matrix $C$ ? Thank you for your attention.
1
vote
0answers
34 views

Let $f$ be a real-valued continuous function on $[0,1]$ which is twice continu-ously differentiable on $(0,1)$. Suppose that $f(0) = f(1) = 0$

Let $f$ be a real-valued continuous function on $[0,1]$ which is twice continu-ously differentiable on $(0,1)$. Suppose that $f(0) = f(1) = 0$ and $f$ satisfies the following equation: $$x^2f''(x) + ...
0
votes
2answers
37 views

What exactly does f'(x)=0 imply from the definition of differentiability?

Let f be a real valued function satisfying $|f (x) −f (a)| ≤ C|x−a|^γ$, for some γ > 0 and C >0. (a) If γ = 1, show that f is continuous at a; (b) If γ > 1, show that f is ...
1
vote
1answer
38 views

Matrix --> Scalar Valued Function: Differentiation

In class, we called a real-valued function from the space of matrices to the reals $f: \mathbb{R}^{m \times n} \rightarrow \mathbb{R}$ differentiable at $\mathbf{X}$ if: $$\lim_{\mathbf{H} \to ...
1
vote
1answer
35 views

Nonstandard example of using chain rule

Find $F'(x)$ if $F(x)=\int_0^x g(s,x+s) \,ds$. Could you please help? I don't know how to start. I suppose I should use the chain rule and the formula $\frac{d}{dx}\,\int_0^x h(s)\,ds=h(x)$ but I ...
2
votes
3answers
40 views

$f(x)=e^x -10x^2$ doesn't vanish in more than three points.

How can I prove that $f(x) = e^x - 10x^2$ doesn't vanish in more than three points. I stuck here I just computed the derivative that is $f'(x) = e^x - 20x$ and then $x= \log(20)+\log(x)$ when ...
3
votes
2answers
50 views

When do we have the formula $f(t)=e^{\lambda t}f(0)+\int_0^te^{\lambda (t-s)}g(s)ds$?

Let $g:\mathbb{R}\to \mathbb{R}$ be a continuous function. Consider the following integral equation $$f(t)=f(0)+\int_0^t\lambda f(s)ds+\int_0^tg(s)ds. \tag{1}$$ Since $g$ is continuous, Thus the ...
0
votes
1answer
18 views

Profit Increase Decrease and Extremes

Im a Survey of Calculus student in LA that needs help with the following problem: A company sells bicycle helmets and the following is their profit function ** P(f)= x^3-12x^2+41x-30 ** P is in ...
1
vote
0answers
25 views

Continuously differentiable positive definite function

Suppose there is a continuously differentiable multivariable scalar function $f$ such that for $\forall {\bf{p}} \in {R^n}$ $f\left( {{\bf{p}},{\bf{q}}} \right) > 0$ $\forall {\bf{q}} \in {R^n} - ...
1
vote
1answer
17 views

negative fraction exponent and division

Quick question on how to handle negative fraction exponents when differentiating: I have this problem to differentiate. $$x^{2/3} + y^{2/3} = 1$$ So my textbook and I both did the first thing the ...
0
votes
1answer
23 views

Partial derevatives of multi variables

Hi I don't understand this problem. $F$ is a function of $r$, right? I can find the partial derivatives $\frac{\partial r}{\partial x}, \frac{\partial r}{\partial y}$, and $\frac{\partial ...
1
vote
1answer
26 views

Derivative of unknown compound function

The problem says: What is $f'(0)$, given that $f\left(\sin x −\frac{\sqrt 3}{2}\right) = f(3x − \pi) + 3x − \pi$, $x \in [−\pi/2, \pi/2]$. So I called $g(x) = \sin x −\dfrac{\sqrt ...
3
votes
1answer
33 views

Derivative of function with respect to $x$ where $x$ is the order of a derivative of another function.

When I learn math I always have lot of thoughts and ideas in my head and some of them are weird. But I came aross a question, which is... also kind of weird. How can a problem like this be solved? Is ...
2
votes
3answers
50 views

Derivatives: If $f(x)= 1/e^x $ then

If $f(x)= 1/e^x $ then $ƒ′(x) = ?$ A: $1/e^x⋅ln(e^x)$ If $ƒ′(x) = e^x$ then $ƒ(x) = ?$ A: $x$ Are my solutions correct?
1
vote
1answer
48 views

Show the Cauchy-Riemann equations hold but f is not differentiable

Let $$f(z)={x^{4/3} y^{5/3}+i\,x^{5/3}y^{4/3}\over x^2+y^2}\text{ if }z\neq0 \text{, and }f(0)=0$$ Show that the Cauchy-Riemann equations hold at $z=0$ but $f$ is not differentiable at $z=0$ ...
0
votes
4answers
52 views

right circular cylinder inscribed in a sphere

Find the dimensions of the right-circular cylinder of greatest vloume that can be inscribed in a sphere with a radius of 6 $in$ I think I need help visualizing, and maybe the solution. I've ...
-1
votes
0answers
29 views

Gradient of a vector [closed]

Matrix $V$ is 200 by 785. Matrix $X$ is 785 by 1. Matrix $W$ is 10 by 201. Matrix $y$ is 10 by 1. First, I do: $ V * X$ Then, I apply $tanh()$ to every element of that resulting matrix. The result ...
-2
votes
1answer
32 views

If f(x) is continuous on $[a,b],$ differentiable on $(a,b)$, and $f'(x)\neq 0\ \forall x\in(a,b),$ then f'(x) is stable. [closed]

If $f(x)$ is a continuous function on $[a,b]$ and differentiable on $(a,b)$, and $f'(x)\neq 0\ \forall x\in(a,b)$ then $f'(x)$ is stable $\left(\text{i.e.},\ f'(x)<0\ \ \text{or}\ \ ...
1
vote
3answers
85 views

Maximum and minimum of of $f(x)=|x-1|+|x-2|+|x-3|$

I am trying to find the maximums or minimums of $$f(x)=|x-1|+|x-2|+|x-3|$$ (if there exist). My attempt: First I compute the derivative and tried to find critical point, i.e, $f'(x) = ...
0
votes
0answers
12 views

Deriving marginal effects in a conditional multinomial logit model

I'm having difficulties deriving the derivative $\frac{\partial P_{ij}}{\partial x_{ih}}=-p_{ij}p_{ih}\beta$ when $P_{ij}=\frac{e^{x_{ij}^{'}}\beta}{\sum_{h=1}^{M}e^{x_{ij}^{'}\beta}}$ I've already ...
0
votes
1answer
40 views

Derivative of log-likelihood cost function with respect to a matrix

Recently, I am learning derivative method to a function and thanks to @hans help, I can solve those which can be expressed by Frobenius product. But for the log-likelihood function, I do not how to ...
-1
votes
1answer
30 views

Differentiation of a trig function [closed]

Please help me to differentiate following equation $$ y=\frac{\cos3x}{\sin2x} $$
0
votes
1answer
14 views

Significance of derivative in finding square free decomposition

If $gcd(f(x),`f(x))=1$ then f(x) is square free. But what is the reason behind taking derivative of f(x)? How one came to this conclusion?
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0answers
28 views

Higher Order Derivative Tests in Multiple Dimensions

To evaluate the minima, maxima, and saddle points of a real function of 2 variables, we use the second derivative test after evaluating the critical points to identify the type of extrema they are. ...
-2
votes
1answer
24 views

Forms of functions in dynamical systems

I wanted to read some introductory material about dynamical systems since I might need a basic understanding of them in a related task. So, as far as I see, in a continuous time dynamical system, we ...
1
vote
0answers
16 views

Proving non-differentiability in a basic sense

I've tried to make this question general enough that it solves other users' questions! Perhaps a solution to this post will explain how one might use a basic definition (see $\mathbb{\S}$) to prove ...
0
votes
0answers
16 views

Question about linearising system with second derivative

I need to linearise a system: $\ddot{x}+4\dot{x}^5+(x^2+1)u=0$. The referenced answer is :$\ddot{x}+0+(0+1)u\approx0$. So, the linearly approximated about $x=0$ is: $\ddot{x}=-u$ I can understand ...
0
votes
0answers
36 views

Do irrational derivative orders exist?

There are many notations for a derivative of $y$ with respect to $x$. Two, most popular are $y'(x)$ or just $y'$ and $\frac{dy}{dx}$. For higher order derivatives, the more consistent notation is ...
4
votes
2answers
57 views

Difference in use between $d$, $\partial$, $\operatorname d$, $\varDelta$ and $D$ for derivatives.

While reading different sources on implicit differentiation (and thereafter differentiation in general), I came across many different "d's" being used for (or similar to) the familiar ...
2
votes
0answers
29 views

Matrix Calculus and Linear Transformations

I'm working on making the jump from differentiating real valued functions ($f: \mathbb{R}^n \rightarrow \mathbb{R}$) and vector valued functions ($g: \mathbb{R}^n \rightarrow \mathbb{R}^m$) to matrix ...
8
votes
1answer
59 views

Prove that $\exists x_0, x_1\in (0,1)$, such that $\frac{f'(x_0)}{x_0}+\frac{f'(x_1)}{x_1^2}=5$

Let $f:[0,1]\to\mathbb{R}$ be a differentiable function, such that $f(0)=0$ and $f(1)=1$. Prove that there exist different $x_0, x_1\in (0,1)$, such that ...
0
votes
2answers
19 views

How would I calculate an derivative with two unknown variables?

I'm learning calculus II. I recently wondered what if I had two unknown variables in an function, and wanted to take an derivative. Let's say there is a function $f(x,y)=2x^3+7y^2$ How would I ...
11
votes
4answers
322 views

Horizontal tangent line of a parametric curve

Suppose $x=t^2,y=t^3$ is a parametric curve. Here's a quote from my textbook: The origin, which corresponds to $t=0$, is a singular point of the parametric curve, because $dx/dt=2t,dy/dt=3t^2$ are ...
-1
votes
1answer
18 views

Differentiating a squared quantity

I was reading through my electromagnetism book where i came across this statement where when we differentiate wrt a squared quantity rather than a single quantity we multiply it by $\frac{1}{2}$. ...
0
votes
0answers
23 views

Prove that $f$ is analytic and derivatives of all order $f_n^{(k)}$ converge to $f^{ (k)}$ uniformly on any compact subset of $G$.

Suppose $f_n$ is analytic in some region $G$ and suppose $f_n$ converges to $f$ uniformly on any compact subset of $G$. Prove that $f$ is analytic and derivatives of all order $f_n^{(k)}$ converge to ...
0
votes
1answer
24 views

Range of this double trigonometric function

What is the range of $$ \sin(\cos(x)). $$ Generally we have various methods. Like differentiation, graphical analysis. So it's range is $[0,1]$. But how to prove it using any known method or ...
3
votes
3answers
72 views

Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$

I was given a task to prove that inequality is true for x>0: $(e^x-1)\ln(1+x) > x^2$. I've tried to use derivatives to show that the $f(x) = (e^x-1)\ln(1+x)-x^2$ is greater than zero, but has never ...
3
votes
2answers
35 views

Number of real roots of $f ' ( x )$

Let $$f(x)=(x-a)(x-b)^3(x-c)^5(x-d)^7 $$ where $a,b,c,d$ are real numbers with $a < b < c < d$ . Thus $ f ( x )$ has $16$ real roots counting multiplicities and among them $4$ are ...
0
votes
1answer
23 views

Calculatin a partial derivative

If we had: and we were to calculate How is it equal to w ?
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vote
2answers
32 views

Find $f(1)$ and $f'(1)$ of $\lim_{h\rightarrow\ 0}\frac{f(1+h)}{h} = 5$

Suppose the function, $f$, is differentiable at $x = 1$. $$\lim_{h\rightarrow\ 0}\frac{f(1+h)}{h} = 5$$ Find a) $f(1)$ and, b) $f'(1)$. I know b) (well at least I think it can) can be found by the ...
0
votes
0answers
7 views

Question conserning the existence and continuity of derivatives of function's shperical mean

I heard a rumor that the claim beneath is true and I'm trying to prove it (or find a counterexample). Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$, $f\in C^k(\mathbb{R}^n)$. Fix $\varepsilon > 0$ ...
1
vote
1answer
55 views

Prove that $f(x)=\sum_{n=1}^{\infty} (\frac{x}{n}-log(1+\frac{x}{n}))$ is continuous and can be differentiated ad infinitum

We have $f:(0,\infty) \rightarrow \mathbb{R}$ defined by infinite series $f(x)=\sum_{n=1}^{\infty} (\frac{x}{n}-log(1+\frac{x}{n}))$ Prove that $f$ is continuous and can be differentiated ...
0
votes
1answer
26 views

Let B be set of all twice differentialbe function $ f(0)=1, f'(0)=-1$ . .. Find supremum of $ {(f''(0):f\in B})$

Let B be set of all twice differentiable function $f$ such that $f: (-1,1) \to (0,\infty)$ and $ f(0)=1, f'(0)=-1$ . We have new function $g(x)$ such that $g(x)=\frac{1}{f(x)}$ and $g(x)$ is convex ...
0
votes
2answers
42 views

derivative of differentiable function [duplicate]

Edited: It is known that if $f$ is differentiable then the derivative function of $f$ is not always continuous. For instance $f(x)=x^2\sin (\frac{1}{x})$ for $x\neq 0$ and $f(0)=0$ if $x=0$. Then ...
0
votes
1answer
62 views

What does $f'(xy)$ mean?

I apologize in advance for the silliness of such question, but what is the meaning of $f'(xy)$ in $yf'(xy) = f'(x)$? Is it the total derivative of $f$ w.r.t $x$? Or it is the derivative w.r.t $xy$?
0
votes
1answer
38 views

$f \in C^1[0,\infty)$ such that $\lim_{x \to \infty} \dfrac {xf(x)}{f'(x)}=2$ ; then for $s<2$ ; $\lim_{x \to \infty}x^{-s}f(x)=\infty$?

Let $f \in C^1[0,\infty)$ be such that $\lim_{x \to \infty} \dfrac {xf(x)}{f'(x)}=2$ ; then is it true that for $s<2$ , $x^{-s}f(x) \to \infty$ as $x \to \infty$ ?
1
vote
1answer
56 views

Rewriting of functional equation $f(xy)=f(x)+f(y)$

Given the following equation: $f(xy)=f(x)+f(y)$ and the fact that $y=x^{-1}$, I've to find how this could became: $f'(x) = f'(1)/x$, where it is said that $f'(x)$ is the total derivative of $f$ ...
6
votes
3answers
366 views

Deriving the Normalization formula for Associated Legendre functions: Stage $1$ of $4$

The question that follows is needed as part of a derivation of the Associated Legendre Functions Normalization Formula: ...
0
votes
0answers
48 views

Different results in differentiation; I don't see the flaw

Might be getting sleepy here, but I cannot find an issue in both methods which yields different results. The function in concern is $f(\theta)=nk \text{log}\theta - (k+1)\sum^n ...