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

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Derivative from Graph

Hi, I am trying to study a bit ahead for my calculus class next year and I came across this question. I was wondering how to find the derivative of the graph without the function. I figured I could ...
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2answers
26 views

Determine the point(s), (if any), at which the graph of the function has a horizontal tangent.

Determine the point(s), (if any), at which the graph of the function has a horizontal tangent. $y(x)= x^4-500x+2$ So I know the first thing to do is find the derivative which is: $y'(x) = 4x^3-500$ ...
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1answer
35 views

Having derivative in some $x_{0}$ implies having it in $U(x_{0})$ [closed]

Let $f$ be continious function in R and it has a derivative in $x_{0}$. Does it have derivative in some $U(x_{0})$?
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2answers
155 views

The right procedure on difficult related rates problems

I'm pretty sure the sample problems my teacher gives to us violate some article of the Geneva convention. I'm in talks with my embassy about that, but in the mean time maybe you guys could look over ...
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1answer
67 views

How to find the roots of the second derivative of $ f(x)=x^2(x − 3)^3$?

I am horrid at factoring and I have to find the inflection points of $ f(x)=x^2(x − 3)^3$. So I to find the inflection points I need to set $f'$ equal to $0$ So I have ...
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1answer
53 views

Why does secant method converge

Assume $f$ is continuous and twice differentiable on $[a,b]$ such that $f'(x)>0$ and $f''(x)>0$, $x \in [a,b]$. If $f(b)>0$ and $f(a)<0$ and I choose $x_0=a$,why are we gauraunteed ...
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2answers
284 views

Find dx/dt given that x=-8, y=9, and dy/dt=5

I have this question on my homework assignment. Assume that x and y are differentiable functions of t. Find dx/dt given that x=-8, y=9, and dy/dt=5. Equation: y^2-x^2=17. There are examples in the ...
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0answers
177 views

Rendering the derivative of composite functions from a graph

I'm on a workbook problem and I want to make sure I'm doing it properly. The problem asks me to find the derivatives of composite functions when given only the graphs of the original functions, here ...
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4answers
77 views

Derivative with respect of a function

i have a function of two variables: $f(\theta,\phi) = \theta \sin(\phi)$ and i have to differentiate $f(\theta,\phi)$ with respect to: $1 - 0.5\theta^2$ That is: ...
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1answer
56 views

Having trouble with differentiating under integral sign

I am sorry if this seems like a dumb question, but I am having trouble in applying differentiation under the integral sign to definite integrals such as this one: $$\int^{1}_{0} ...
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1answer
78 views

Partial derivative and derivative.

I want to show that if $f:\mathbb{R}^n\to \mathbb{R}$ and $df_a$ is the derivative of the function at $a$ then $df_a(v)=\displaystyle\frac{\partial f}{\partial v}(a)$. I saw a few proofs of this ...
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1answer
55 views

Solving ODE with matrices

I have an equation in ODE $M{'}(x)= M(x)*A(x)$. Issue here is $A(x) = C_1+C_2* x $ where $C_1,C_2 $ has dimension $3 \times 3$. And x is a scalar variable Doubt What is M(x)? Can any one give ...
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2answers
108 views

Matrix exponential Differentiation

We have the equation $e^X = \sum_{k=0}^\infty{1 \over k!}X^k.$, where X is a matrix of dimension $3 \times 3$ . Now I have a function $f(x)=C_1x+C_2*\frac{x^2}{2} $ where $C_1,C_2,f(x)$ has ...
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0answers
40 views

Derivatives, help please.

${E = - \cfrac{\sum\limits_{c}^C \log P(C)Y}{N}}$ where ${\log P(C) = I_c - \log \sum\limits_{c}^C \exp\left(I_c - \max_c\left(I_c\right) \right)}$ ${I_c = WO}$ ${W}$ is ${C \times N.Hid}$ weight ...
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2answers
66 views

Calculus Implicit Differentiation and Concavity

Consider the relation $4x^2 - y^2 = -2$ (a) Use implicit differentiation to calculate $dy/dx$ and find all critical points of the curve. (b) Calculate the second derivative and determine the ...
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191 views

Proof of application of Mean Value Theorem

Two bicyclists begin a race at 8:00AM. They both finish the race 2 hours and 15 minutes later. Prove/explain that at some point during the race, the bicyclists are traveling at the same velocity. So ...
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3answers
319 views

How to differentiate $\lim\limits_{n\to\infty}\underbrace{x^{x^{x^{…}}}}_{n\text{ times}}$? [duplicate]

Let $$f(x)=\lim\limits_{n\to\infty}\underbrace{x^{x^{x^{...}}}}_{n\text{ times}}$$ Is it possible to find $f'(x)$. If yes, please show all steps.
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1answer
75 views

Stuck trying to prove that $e^{-x^{-2}}$ is $C^{\infty}$ [duplicate]

This is Spivak's Calculus on Manifolds ex. 2-25, he says Define $f:\mathbb{R}\to \mathbb{R}$ by $f(x) = \left\lbrace \begin{array}{l} e^{-x^{-2}} &\text{ if } x \neq 0\\ 0 ...
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2answers
51 views

Problem related to Mean Value Theorem

I found out a question that I can't figure out a way to solve it. Plz can anyone help me. Question is, Prove that $\exists\,C\in(0,\pi/4)\,\mathrm{s.t.}\,\tan(\pi/4+C)=3/C$ I know this should be ...
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4answers
103 views

Implicit Derivative approaches

Sorry for my excessive verboseness... Here's the equation as given: $$x = 10 + \sqrt{x^2 + y^2}$$ Here are my direct implicit steps without modifying original equation: $$\eqalign{ \dfrac{\mathrm ...
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0answers
53 views

Solving system of differential equations

I have a system of differential equation to solve. Any suggestions regarding closed form or numerical method is welcome with great respect. This equation is from dynamic equation of a curve. Let us ...
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1answer
24 views

Find the equation of a line tangent at a specific point

I have to find an equation for the line tangent to the graph of $\large\frac {\sqrt{x}}{6x+5}$ at the point $(4,f(4))$, and write it out in the form of $y=mx+b$ Using the quotient rule I get.. ...
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3answers
241 views

surjective, but not injective linear transformation

$T$ is a transformation from the set of polynomials on $t$ to the set of polynomials on $t$. So, the input to $T$ should be a polynomial, and the output should be some other polynomial. Two common ...
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1answer
54 views

Differentiating Integrals

This problem appears as example 2d of Chapter 5 in "A First Course in Probability - Ross, 8th ed." Suppose that if you are s minutes early for an appointment, then you incur the cost cs, and if you ...
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3answers
81 views

Using the Chain Rule to prove trig derivatives

I'm having trouble with this problem, I'm not sure how to tackle it and I was wondering if somebody could set me on the right path. The problem is as follows: Use the Chain Rule to show that if ...
2
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1answer
81 views

Finding $\dfrac{d^nx}{dy^n}$

If $y$ is a function of $x$, then what is the relation between $\dfrac{d^nx}{dy^n}$ and $\dfrac{d^ny}{dx^n}$? If we were to talk about $\dfrac{dy}{dx}$ and $\dfrac{dx}{dy}$, then they both are ...
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1answer
44 views

Aftermath of Cauchy's mean value theorem

Let $f(x)$ be a real-valued function defined on a closed interval [a, b], differentiable on the open interval (a, b) $n-1$ times. $x_0$ belongs to [a, b]. Suppose that we ...
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1answer
58 views

Uniform convergence result in proof of second-derivative formula

This is a fairly basic analysis question. Consider a continuous function $f: \mathbb{R} \to \mathbb{R}$ which is twice differentiable at a point $x$. If necessary, also assume that $f \in ...
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3answers
120 views

What do we lose by differentiating without using the rules of differential calculus?

I learned differential calculus and its rules (quocient, chain, etc) and I got curious about one thing: What do we lose by not using these rules when differentiating? Obviously I've noted some utility ...
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1answer
35 views

Question about Peano form of the remainder

Let $f(x)$ be a real-valued function defined on a closed interval [a, b], differentiable on the open interval (a, b) $n-1$ times. $x_0$ belongs to [a, b]. Suppose that we ...
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2answers
77 views

Find the derivative of $y=\cos(x) - 2\sin(x),$ when the gradient is $1$

I need to find the smallest positive value of $x$ for which the gradient of the curve has value 1. For this equation: $$ y =\cos(x)-2\sin(x) $$ The answer is 2.5c grad. The following is my ...
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0answers
56 views

Is this connected relation?

My task is to check if this is preference relation (connected and transitivited) $$ f \succeq g \Leftrightarrow \forall x\in [0,1] f'(x) \leq g'(x) $$ My solution is: that this relation is not ...
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2answers
66 views

If $f'(x)\cdot x$ goes to zero then $f(2x)-f(x)$ is bounded.

Let $g:\mathbb R^m\to\mathbb R^n$ be defined by $g(x)=f(2x)-f(x)$ where $f:\mathbb{R}^m\to\mathbb{R}^n$ is a given differentiable function. The problem is to prove that if $\lim_{|x|\to\infty} ...
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3answers
68 views

Trouble finding the derivative of an expression

I could use your help. I've spent over 20 minutes on this problem and my inability to solve it has my questioning my calculus skills. If someone could show me where I messed up and walk me through the ...
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1answer
61 views

Anti derivative notation [duplicate]

$F$ is an anti derivative of $f$. $$\int f(x) dx = F(x)+C$$ Can you tell me why there is '$dx$' in the LHS?
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2answers
68 views

Proving double derivatives with the chain rule (I think?)

Hey StackExchange I'm having trouble understating where to start with this problem, I'm supposed to prove something about double derivatives and the chain rule but I'm having trouble understanding ...
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1answer
23 views

Discovering the derivatives of functions combined with trig values.

Hey StackExchange I have a problem that I don't really understand and I could use some hints for starting it. Suppose $m(\frac{\pi}{3}) = 4$ and $ m'(\frac{\pi}{3}) = -2$, and let $g(x) = m(x)\sin x$ ...
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1answer
27 views

How rigorous is multiplying both sides of an eqaution for the differential of a function?

I have to solve this equation: $$ -C_0 f + \frac{1}{2}f^2 +\frac{d^2 f}{d X^2}=A $$ where $C_0$ and $A$ are two real nonzero constant; $f:\mathcal{R}\to \mathcal{R}$ I have seen that the person who ...
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1answer
27 views

Second derivative test of a function of two variables

From the following relation: How can we conclude the following rules: (i) Minima if both $f_{xx}$ and $f_{yy}$ are positive and $(f_{xy})^2 < f_{xx} f_{yy}$, (ii) Maxima if both $f_{xx}$ and ...
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0answers
121 views

Sign of the derivatives of a simple function

Consider the function $f(x)=x^b(1-x)^{1-b}$ defined on $[0,1]$, with $0 < b <1$. How can we prove that the even derivatives $f^{(2k)}$ have a constant sign on $(0,1)$? One can show that this ...
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2answers
391 views

Is it differentiable?

Let us consider the function $$ f(x)= \begin{cases} x^2\sin {\dfrac{\pi}{x}} & x \neq 0\\ 0 & x=0 \end{cases} $$ We want to check its differentiability at $x=0$. By ...
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50 views

Find the absolute maximum and absolute minimum values of f on the given interval

Find the absolute maximum and absolute minimum values of f on the given interval. $f(t) = t\sqrt{9 - t^2}$ on the interval $[-1,3]$. So $f'(x)=\frac{t}{2\sqrt{9-t^2}}+t\sqrt{9-t^2}$ and that is as far ...
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1answer
36 views

Two definitions for a smooth curve equal.

I've encountered these two definitions: 1. $\gamma\colon [a,b]\longrightarrow\mathbb{R^3}$ is smooth if all three derivatives exist and $\gamma^{\prime}(t) \neq 0 \;\; \forall t \in [a,b]$ ...
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42 views

Prove (non)differentiability in piecewise functions

I'm looking for some help on proving that this function is not differentiable at a specific value. My first instinct is to approach the limit of the value from positive and negative, but that doesn't ...
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2answers
78 views

Gradients and functions on matrices

Given a twice differentiable $f: \Bbb R \to \Bbb R$, with continuous second order derivative. We define $$F(x) = \sum_{i=1}^{m}f(x_i)$$ and $$L(x) = \sum_{i=1}^{m}f( \langle a_i, x \rangle+ b_i),$$ ...
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3answers
55 views

Factoring when differentiating expressions

I'm having trouble with differentiating a expression. I do it one way, wolfram alpha does it another. Let me show you what I mean. The original expression is this: $$\frac{1}{2u^3}$$ I start by ...
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2answers
49 views

About matrix derivative

Suppose $A$ is a matrix with order n*n. we have the following equity but I don't know why. $f(x)=\frac{1}{2}x^TAx-b^Tx$. then $f'(x)=\frac{1}{2}A^Tx+\frac{1}{2}Ax-b$ Is there any rule like scalar ...
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1answer
40 views

$\displaystyle k^{th}$ derivative of a Gaussian function with zero mean

The gaussian function is: $$f(x,\mu,\sigma)=\dfrac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\dfrac{(x-\mu)^2}{\sigma^2}\right)$$ Putting $\mu=0$, we can get the $\displaystyle k^{th}$ derivative of this ...
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2answers
108 views

What is the best way to find the derivative of binomials to a power? ((x+x^{-1})^3)'

I came to a problem on my homework and I want to know the best way to solve it. We are doing derivatives in Calculus. I've got the following: $$H(x)=(x+x^{-1})^3$$ $$H'(x)=((x+x^{-1})^3)'$$ I am ...
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2answers
40 views

Find the Derivative of fraction

I can't find out what I'm doing wrong again... $$f(x)=\frac{x^2+4x+3}{\sqrt{x}}$$ $$f(x)=\frac{x^2}{\sqrt{x}}+\frac{4x}{\sqrt{x}}+\frac{3}{\sqrt{x}}$$ $$f(x)=x^2(x^{9-1/2}) + ...