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

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95 views

What is this symbol called and what is it's use?

I have been seeing this symbol ever since I started university and I am finding it hard to Google-fu what it is. Can someone tell me the name of it and hopefully the function of it as well? It is ...
2
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1answer
68 views

Derivatives of second order

Consider a real function $f$ of one variable. Suppose the second order derivative exists. To find the second order derivative of $f$, I usually derivate $f$ two times. I start with $f$, and derivate ...
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2answers
34 views

Partial fraction when $N^r$ and $D^r$ are quadratic and cubic polynomials

I need to find the nth derivative of the following function $$y=\frac {x^2+4x+1}{x^3+2x^2-x-2}$$ The trouble is I don't know how to break a fraction like the above one. How do I break it into partial ...
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2answers
224 views

Let $f,g$ be differentiable with $f(0)=g(0)$ and $f'(x)<g'(x)$. Prove that $f(x)<g(x)$.

Let $f,g:\mathbb{R} \rightarrow \mathbb{R}$ be differentiable with $f(0)=g(0)$ and $f'(x) < g'(x)$ for all $x$ belonging to the set of real numbers. Prove that $f(x)<g(x)$ for all $x>0$. Any ...
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0answers
39 views

Could someone please help me solve this problem?

I have a term: $$\int_0^x\left(\int_y^x e^{-(c+d)(z-y)}e^{-ky} \,dz\right) \,dy.$$ I want to differentiate this with respect to $x$. $c$ and $d$ are constants, and I have tried using Leibniz's ...
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4answers
85 views

derivative of an integral from 0 to x when x is negative?

Given a function $$F(x) = \int_0^x \frac{t + 8}{t^3 - 9}dt,$$ is $F'(x)$ different when $x<0$, when $x=0$ and when $x>0$? When $x<0$, is $$F'(x) = - \frac{x + 8}{x^3 - 9}$$ ... since you ...
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2answers
44 views

differentiation - tangent to the curve $f(x) = (2x-1)(x+1)$

Find the equations of the tangents to the curve $f(x) = (2x-1)(x+1)$ at the points where the curve cut the x-axis. find the points of intersection of these tangents.
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2answers
85 views

Derivative of a map involving the matrix inverse

I have $f: U\rightarrow \mathbb{R}$, $f(X):=\operatorname{tr}(X^{-1})$, $U$ contains all matrices $X$, which are positive definite and symmetric. I want to show that $f$ is differentiable on $U$. To ...
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2answers
50 views

Differentiating under the integral sign chain rule

Can someone explain to me why $$ \frac{\partial}{\partial x}\int_{0}^{x\nu}u^{c - 1}{\rm e}^{-u/2}\,{\rm d}u = \left(\nu x\right)^{c - 1}{\rm e}^{-\nu x/2}\,\nu\quad {\large ?} $$ I know it has to do ...
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2answers
408 views

Find the dimensions of a cylinder of given volume V if its surface area is a minimum.

The following is the question : Find the dimensions of a cylinder of given volume V if its surface area is a minimum. The cylinder has a closed top and bottom. 2 formula : (1) $V=r^2\pi ...
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1answer
16 views

the following formula shows the relationship between the amount of energy (E) released and the richter number. M = 2/3log10(E/0.007)

E is measured in kWh hours. If the average household uses 247 kWh hours per month, how many months would the energy generated released by an earthquake measuring 7.7 on the richterscale power 4.8 ...
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4answers
109 views

What is $\frac{d(\arctan(x))}{dx}$?

Let $v= \arctan{x}$. Now I want to find $\frac{dv}{dx}$. My method is this: Rearranging yields $\tan(v) = x$ and so $dx = \sec^2(v)dv$. How do I simplify from here? Of course I could do something like ...
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0answers
49 views

Taylor expansion of $f(x+y)$

I have the following question: Let $\psi(t)$ be a function with bounded derivatives of any order on R. Find Taylor’s expansion for the two variable function $f (x, y) = \psi(x + y)$ at $a = (0, 1)$. ...
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0answers
29 views

Euler operator- (variational derivative) for linear dispersive wave

I was analyzing the Euler operator, also known as variational derivative, it’s got a nice property in obtaining the conserved densities of partial differential equation, I tried to verify its effect ...
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2answers
126 views

Finding the tangent line(s) to a curve

first time poster so sorry if I'm doing something wrong. "Consider the closed curve in the xy-plane given by $2x^2 - xy + y^3 + x = 9$. Find equation(s) of all tangent lines to the curve at $y = ...
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1answer
41 views

Related rates question?

I am trying to solve the following question but I am not sure how to approach it. I know that I have to get the derivative of s but how do I get the rate at which sales are currently changing? A ...
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1answer
30 views

How do I differentiate this integral within an integral with respect to t? $b(t) =\int_0^t (k(e^{-(\mu+\gamma)x})\int_0^t e^{-ky} dy)dx$

I have the following equation and I need to differentiate it with respect to t (to get $b'(t)$). $$b(t) =\int_0^t (k(e^{-(\mu+\gamma)x})\int_0^t e^{-ky} dy)dx.$$ I am confused as to how to deal ...
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2answers
62 views

What is meant when $f:[a,b] \to \mathbb R$ is said to be differentiable?

Sometimes I see an exercise like this: Let $f:[a,b] \to \mathbb R$ be differentiable. (A few more givens here.) Show that $f'$ has such-and-such property. What is usually meant by that? Should ...
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1answer
37 views

How do you prove $e^{-a}=a$ without using graphs?

We're doing a section on limits, continuity, and differentiation in my Advanced Calculus class, and I am at a loss for how to prove this...
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2answers
95 views

The derivative of $\tanh x$

I'm trying to calculate the derivative of $\displaystyle\tanh h = \frac{e^h-e^{-h}}{e^h+e^{-h}}$. Could someone verify if I got it right or not, if I forgot something etc. Here goes my try: ...
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4answers
166 views

Why does $( \operatorname e^x)' = \operatorname e^x?$ [duplicate]

It's known the the derivative of exponential function $a^x$ is $xa^{x-1}$. If I play $e$ as $a$, we'll get $(a^x = \operatorname e^x)' = x \operatorname e^{x-1}$. Why does $(\operatorname e^x)' = ...
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2answers
93 views

Does it imply $f$ is differentiable on $\mathbb R?$

$f:\mathbb R\to\mathbb R$ be such that $\forall~x\in\mathbb R$$$ \lim_{h\to0}\dfrac{f(x+h)-f(x-h)}{h}$$exist. Does it imply $f$ is differentiable on $\mathbb R?$
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3answers
74 views

How to get the derivative of $(\ln(x))^{\sec(x)}$?

How do you get the derivative of $(\ln(x))^{\sec(x)}$? I know that the derivative of $\ln(x)$ is $\frac 1x$ but what happens when you take it to an exponent of $\sec(x)$?
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1answer
32 views

Elementary differentiation question on derivation of p.d.f. of function of random variable

Let $G(y) = \Pr(Y \le y) = 1 - F(\frac{1}{y})$. Then apply the chain rule (assuming $y \ne 0$ and $F(x)$ is differentiable at $x = 1/y$) and we have $$g(y) = \frac {d\ G(y)}{dy} = \frac{-d\ F(x)}{dx} ...
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1answer
53 views

How to make this change of variables?

"Show that if we introduce the independent variable $x = \sqrt{\frac{z}{L}}$ then the equation $zZ''(z) + Z'(z) + v^2Z(z)=0$ becomes $Z''(x) + \frac{1}{x} Z'(x) +4v^2LZ(x)=0$ for $0<x<1$. So ...
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1answer
49 views

On derivatives…

I have a quick question here. I hope someone can help. I haven't done calculus for a long time so I seem to missed out on details. If $x=g^{-1}(y)$ and $g$ is monotonic and is differentiable for all ...
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5answers
97 views

Can $f(x)>g(x)$ be implied from $\frac{df(x)}{dx}\gt \frac{dg(x)}{dx}$?

I am new to functions. My question is Can $f(x)>g(x)$ be implied from $\frac{df(x)}{dx}\gt \frac{dg(x)}{dx}$?
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2answers
63 views

How to find the derivative of $\left(\frac{x-2}{x+2}\right)^{1/2}$ [closed]

Does anybody know how to solve this? How do I approach it? $$\ f(x)=\sqrt{\frac{x-2}{x+2}} $$ $$\ f'(x)=? $$
0
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1answer
168 views

Maximizing cross sectional area of trapezoid

The task is to fold a piece of sheet metal that measures 60 cm across in such a way as to form a trapezoidal "gutter" (a trough for carrying rainwater) with the maximum possible cross-sectional area. ...
2
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1answer
38 views

How do I find a function to minimize another function?

I am given to constants $b, n \in \mathbb{N}$. The task is to find a function $r(b,n)$ such that $\text{range}(r)=[1,b]$ and the value of $\frac{b}{r(b,n)}(n+2^{r(b,n)})$ is minimal. Do I have to ...
2
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0answers
792 views

Strictly monotonic increasing function

Suppose that $f$ is continuously differentiable on $[a,b]$ and $f'(x) > 0$ for all $x$. Prove that $f$ is strictly monotonic increasing on $[a,b]$; that is, if $x<y$, then $f(x) < f(y)$. ...
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1answer
51 views

Prove that the function f defined by$f(x)= x^r cos(\frac{1}{x})$ for $x\neq 0$ and $f(0)=0$ is differentiable at $0$ if $r=2$

Prove that the function $f$ defined by $f(x)= x^r cos(\frac{1}{cos})$ for $x\neq 0$ and $f(0)=0$ is differentiable at $0$ if $r=2$ and not differentiable at $0$ if $r=1$
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1answer
55 views

On differentiating an integral with respect to a function

Let $f,g:\mathbb{R}^n \rightarrow \mathbb{R}$, and let $$ Q = \int \! g(\mathbf{x})f(\mathbf{x}) \, \mathrm{d}\mathbf{x} $$ What is the result of the following differentiation? $$ ...
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3answers
716 views

How to find the derivative of $(2x+5)^3(3x-1)^4$

How to find a derivative of the following function? $$\ f(x)=(2x+5)^{3} (3x-1)^{4}$$ So I used: $$(fg)'= f'g + fg'$$ and $$(f(g(x)))'= f'(g(x)) + g'(x)$$ Then I got: $$ f(x)= 6(2x+5)^{2} + ...
3
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1answer
54 views

Prove there are $\xi,\eta$with $f'(\xi)f'(\eta)=1$

Let $f:[0,1]\to\mathbb{R}$ be continuous with $f(0)=0,f(1)=1$ and $f$ is differentiable on $(0,1)$. Show that there are distinct $\xi,\eta\in(0,1)$ so that $f'(\xi)f'(\eta)=1$. I think this requires ...
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0answers
27 views

Hessian of a conic function

i got a conic System: $Ax =b, x\in C$, where $A\in\mathbb{R}^{m\times n}, b\in\mathbb{R}^m$ and C is the cone of the $n\times n$ positive semidefinite matrices, so ...
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3answers
48 views

Question about differentiation and equalities and integrals

Question ;Let's suppose I have function f(x) and function u(x) Now if $\frac{d}{dx}f(x) = q$ and $\frac{d}{dx} u(x)=q$ then this means $u(x)=f(x)$?" But if I do integral of q would I get $f(x)$ or ...
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2answers
246 views

Clarification on Rules Differentiation and First Principles Derivatives

My grade 11 class has just started differential calculus, the one area seemingly glazed over in our book. We have covered some simple rules of differentiation, like f(x) = n x^(n-1), and have applied ...
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0answers
46 views

How to show $(f\circ\gamma)_{\star q}=\mathcal{J}(f\circ \gamma)_q $

Let $ \gamma :R\rightarrow R^n$ and $f:R^n\rightarrow R$. If $q=0$; show that $$(f\circ\gamma)_{\star q}=\mathcal{J}(f\circ \gamma)_q $$ My Attempt: We know that $$\mathcal{J}(f\circ ...
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2answers
116 views

Notation regarding different derivatives

I am currently reading up on partial derivatives and differentials in general. And there are a few points that seem unlcear to me (notation-wise). For example, if $f:\mathbb R\to\mathbb R,x\mapsto ...
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2answers
148 views

How to prove the equation $\cos x=2x$ has only one solution? [closed]

Show that the equation $\cos x=2x$ has only one solution, $x\in\mathbb{R}$.
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1answer
48 views

Extrema problem

Task: You have a square and you cut out it's corners in shape of a square, so you get a box. The task is to calculate $a'$ of a square (you cut out) if the volume of a box is maximum. I don't ...
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3answers
70 views

If $f $ is differentiable at $c$, then $\lim\limits_{x\to c} \frac{xf(c)-cf(x)}{x-c}=f(c)-cf'(c)$

Let $f \colon \mathbb{R} \to \mathbb{R}$ be differentiable at $c \in \mathbb{R}$. Prove that $$\lim_{x\to c} \frac{xf(c)-cf(x)}{x-c}=f(c)-cf'(c).$$ This is from a first course real analysis class ...
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1answer
47 views

Derivative Proofs - (c) Prove that the following facts are true about $s$ if $s(t) = (a/2)t ^2$

Hi guys I'm really having trouble with (b) and (c). I did question (a) so thats outta the way but I'm really stuck on these two. I've been working on them for the whole weekend and can't seem to ...
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4answers
194 views

question about the limit $\lim_{h\to0}\frac{\arcsin(x+h)-\arcsin(x)}{h}$

Because $\sin'(x)=\cos(x)$ we can prove that $\arcsin'(x)=\frac{1}{\sqrt{1-x^2}}$. but, by definition we have $$\arcsin'(x)=\lim_{h\to0}\frac{\arcsin(x+h)-\arcsin(x)}{h}\tag{1}$$ therefore, ...
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2answers
36 views

Can you please proof that f is constant by computing the derivative f? [duplicate]

Suppose that $│f(x) - f(y)│≤│x-y│^2$ for all $x,y \in \mathbb{R}$ Proof that $f$ is constant by computing the derivative $f$
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1answer
42 views

Can someone example and give an example?

Given an example of a function f such that lim f(x) when x approaches infinitive exists, but lim derivative of f(x) when x approaches infinitive does not exist.
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1answer
171 views

Derivative word problem - $s''(t) = a$, i.e the acceleration is constant and $(1) [s'(t)]^2 = 2as(t)$

Hi Guys I'm having trouble with $\eqref a$ and a lot more with $\eqref b$. I'm assuming for $\eqref a$ that we must first show why $c$ was switched $c$ to $\frac{a}{2}$. However after that I can't ...
3
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3answers
59 views

Converse of interchanging order for derivatives

We know that for a twice-differentiable function $f$, $$\dfrac{\partial}{\partial x}\dfrac{\partial}{\partial y}f(x,y)=\dfrac{\partial}{\partial y}\dfrac{\partial}{\partial x}f(x,y).$$ Suppose there ...
0
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0answers
27 views

Problem with with derivative of integral

I know about Libnitz rule but I don't know how we can proof this derivation : The second function 'f' is derivation of 'F' function on 'z'. This is not a Homework.It is just a part of a question ...