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

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Polynomial or Exponential

PROBLEM: Let $f(x)$ be a polynomial function. It is known that for every $x$: $$ f'(x) \leq f(x) $$ Prove/disprove: For every $x$: $$ f(x) \geq 0 $$ MY INTUITION: Suppose by contradiction that ...
5
votes
1answer
75 views

Differentiation under integral sign help

Question is: If $$f(a)= \int_0^\infty e^{-t^2}\cdot \cos(at)~dt$$ then I have to show that $f'(a)=-\dfrac{a}{2}\cdot f(a)$. I know that ...
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1answer
37 views

Largest class of functions where derivatives and products commute

What is the largest class of everywhere differntiable real functions of one variable such that the product of the derivatives is the derivative of the product? Certainly the constant functions satisfy ...
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2answers
82 views

Prove that if f '(x) ≠ 0 for all $x\in\mathbb{R}$. , then f is one-to-one. Also, give an example to show the converse of this is false.

Hi guys I have run into another dead end! This is a practice problem for my exam review in my first year calculus class. Any help would be great. Thanks for all your help! Let $f: \Bbb R \to \Bbb R$ ...
3
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2answers
96 views

Show that there is $\eta\in(0,1)$ with $f''(\eta)=f(\eta)$

Suppose $f:[0,1]\to\mathbb{R}$ is twice differentiable with $$\lim_{x\to0^+}\frac{f(x)}{x}=1,\lim_{x\to 1^-}\frac{f(x)}{x-1}=2$$ Show that there is $\eta\in(0,1)$ s.t. $f''(\eta)=f(\eta)$. It is ...
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0answers
30 views

Example regarding the difference quotient

Give an example of a differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $f^\prime(0) = 1$, but there are no points $x,z \in \mathbb{R}$ for which $\frac{f(x)-f(z)}{x-z}=1$ and $x \neq ...
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2answers
70 views

Show that f is differentiable at every point in R, and find the derivative f'

I have been stuck in this problem for a while, its practice question for my exam in my real analysis calculus class. Any help would be great! Thank you! Define $f: \Bbb R \to \Bbb R$ by $$f(x) = ...
2
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1answer
72 views

Question about continuous differentiability

Consider the function $f(x,y) = \frac{x}{1+\sqrt{x^2+y^2}}$. Its derivative with respect to $x$ can be calculated to be $\frac{1 + \frac{y^2}{\sqrt{x^2 + y^2}}}{1 + x^2 + y^2 + 2 \sqrt{x^2 + y^2}}$. ...
1
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1answer
41 views

Derivative of exp with definition of differentiability

Prove with the definition of differentiability that $\exp(z)$ is differentiable in $\mathbb C$ and $(\exp(z))' = \exp(z)$ for all $z \in \mathbb C.$ I tried: \begin{align*} \frac{\exp(z+h) - ...
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1answer
55 views

Complex differentiability equivalent to linear approximation

Let $G \subset \mathbb C$ be an open set and $f: G \to \mathbb C$ a complex function on $G$. Prove that the function $f$ is complex differentiable at a point $z \in G$ if and only if there exists a ...
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1answer
110 views

How to prove that a complex power series is differentiable

I am always using the following result but I do not know why it is true. So: How to prove the following statement: Suppose the complex power series $\sum_{n = 0}^\infty a_n(z-z_0)^n$ has radius of ...
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1answer
24 views

A question on differential equation.

If we consider the equation $x+y+a=0$ (Equation $1$) , we get $1+dy/dx+0=0$ (Equation $2$) as the solution, if we differentiate. We have the reason for writing differentiation of $y$ as $dy/dx$ ...
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votes
2answers
50 views

Differentiate with respect to $x$

Differentiate with respect to $x$ $$f(x)=\sqrt[3]{x^2}-4+\dfrac{8}{x^{2/3}}$$ Solution(is it correct having difficulties with the fractions): $$=x^{2/3}-4+8x^{-2/3}$$ $$=\dfrac{2}{3}x^{-1/3} - ...
5
votes
5answers
119 views

Finding the derivative of $2^{x}$ from first terms?

I was trying to understand why $e^{x}$ is special by finding the derivatives of other exponential functions and comparing the results. So I tried ${\rm f}\left(x\right) \equiv 2^{x}$, but now I'm ...
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2answers
209 views

Differentiation, from first principles

I am having problems with this question, it would be wonderful if someone can help. Given that $f(x)= x^2 + x - 3$ 1) Find $f(x + h)$ 2) Then express $f(x+h)-f(x)$ in its simplest form 3) Deduce ...
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4answers
210 views

Let f: R --> R be a differentiable function satisfying f'(x) = f(x) for all x E R. Show that f(x) = ce^x for some constant c E R.

Hi guys I have been stuck on this question for while now, its a practice problem for my first year real analysis class and I have an exam coming up soon, so I really need your help! Something is ...
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3answers
100 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
votes
1answer
89 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
39 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 ...
2
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1answer
273 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
42 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
133 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
52 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
97 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
65 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
1k 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 ...
3
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4answers
116 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 ...
0
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0answers
53 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
33 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
334 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
43 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
36 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 ...
0
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2answers
65 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
124 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
172 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
95 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 $\mathbb R?$
2
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3answers
79 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
36 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
54 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 ...
0
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1answer
57 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 ...
0
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5answers
100 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
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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? $$\ f(x)=\sqrt{\frac{x-2}{x+2}} $$ $$\ f'(x)=? $$
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1answer
292 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. ...
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1answer
43 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
1k 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
53 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
58 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
1k 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} + ...