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

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3answers
36 views

Equation of line normal to $y = x^3 -2x^2$ at $x=0$

Find the equation of normal to the curve $y =x^3 - 2x^2$ at $x= 0.$ Find the co-ordinates of the point of intersection of the normal and the line $y = 4.$ I differentiated the equation with respect ...
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2answers
50 views

Studying the differentiability of a function at a point $(a_{1},a_{2})$

I have a function $\ f: \mathbb{R}^2 \to \mathbb{R} $ to study: 1) It's continuity at the point $(a_1,a_2)$. 2) The partial derivative exists at $(a_1,a_2)$? 3) Are the partial derivatives ...
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0answers
25 views

Where do the minus sign and Laplacian come from in this derivative?

I want to find this functional derivative: $$\dfrac{\delta \int d^d x'[\nabla_{x'} \phi(\vec{x}')]^2}{\delta \phi(\vec{x})} = \int d^d x' \left(\dfrac{\delta \nabla_{x'} \phi(\vec{x}')}{\delta ...
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1answer
28 views

Minimizing an open box (Calc I)

A rectangular container with an open top is to have a volume of $12 \;\text{m}^3$. The length of its base is twice the width. Material for the base costs (in dollars) 10/$\text{m}^2$. Material for ...
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0answers
38 views

Taking derivative of a partial bell polynomial?

I am trying to prove a statement that involves me taking the derivative of a bell polynomial. Is there an elementary way to express: $$ \frac{d}{dx}[ B_{n,k}(x_1,x_2,....,x_{n-k+1})] $$ Where you ...
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1answer
54 views

Word Problem With Related Rates about Commodity Production

This week, a factory is producing 50 units of a particular commodity, and the amount being produced is increasing at the rate of two units per week. If $C(x)$ is the total cost of producing x units, ...
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1answer
98 views

Partial Derivative Math Homework Help

The attendance (denoted by the variable F , measured in thousands of fans) at a blue Jays home game is approximated by F = 150W^(1/3)P^(2/3) Where W is the fraction of the games they have won so far ...
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0answers
27 views

Derivative of a Decreasing Function

Show that if $C(K,T)$ is a differentiable function of $K$, then the derivative of $C(K,T)$ must lie between between minus one and zero. I have to use the following theorem: $C(K,T)$ is a decreasing ...
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4answers
89 views

If $f$ is a continuous odd function. Prove that if $f$ is differentiable at $0$, then there is a continuous even function $g$ such that $f(x) = xg(x)$

I'm working backwards to see if I can find the $g$, however, when I take the derivative of $xg(x)$ I have $f'(x) = g(x) + xg(x)'$ at $0$, then it will always ends up with $0$. Then I have no idea how ...
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1answer
16 views

Let $f:\Bbb{R}^2\to\Bbb{R}$, with $f(x,y)=\frac{1}{6}(3xy^2-4x^2y+y^3+10)$. Find the equation at the tangent plan of f at the point $(3,2)$.

I've answered this question and I got: $f_x(a,b)=-6$ $f_y(a,b)=2$ $f(a,b)=-3$ and my answer is: $-6x+2y+9$ Just wanted to know whether I am on the right tracks here? The second part to this ...
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1answer
63 views

Need hints/resources to solve this system of equations (seems like a loop)

$\begin{cases} x=1+\ln y \\ y=1+\ln z \\ z=1+\ln x \end{cases}$ I've tried taking the derivative with respect to each variable on the RHS, i.e. ...
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1answer
49 views

Why does Continuous Partial Differentiability Imply Total Differentiability?

Let $f: \mathbb{R}^d \to \mathbb{R}$ be such that the partial derivatives $\frac{\partial f}{\partial x_i}:\mathbb{R}^d \to \mathbb{R}$ exist everywhere and are continuous. Then show that $f$ is ...
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1answer
62 views

Finding derivative of the inverse without the inverse

We are given a function $$f(x)=4\arcsin(\sqrt{x})+2\arcsin(\sqrt{1-x})$$ The derivative of $f$ is: $$f'(x)=\frac{1}{\sqrt{x-x^2}}$$ I would like to find the maximum value of $f^{-1}$. I think I have a ...
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1answer
22 views

A spaceship is traveling (left to right) along the curve y=3cosx.

An object is released from the spaceship at x= pi/3 and travels along a line tangent to the graph of y=3cosx towards the x-axis. a) At what point x will the object strike the x axis? b) At what ...
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3answers
47 views

Find any function $f(x)$ such that $f^{(n)}(0)=\dfrac{n!}{2^n}$.

Find any function $f(x)$ such that $f^{(n)}(0)=\dfrac{n!}{2^n}$. I tried $f(x)=e^{\dfrac{x}2}$, but then $f^{(n)}(0)=\dfrac1{2^n}$. Is there an easy way to find $f(x)$?
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4answers
60 views

Differentiate $\tan(xy)= y+2$

Here is what I did: $$\tan(xy)=y+2$$ $$(xy')(y)\sec^2(xy)=y'$$ Now I'm stuck on simplifying this. How do I get all the y's on one side and divide?
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1answer
46 views

If $f''(x_0)$ exists then $\lim_{x \to x_0} \frac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0)$

Prove: if $f''(x_0)$ exists then $\lim\limits_{x \rightarrow x_0} \dfrac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0)$. I'm not exactly sure how Taylor's theorem fits into all this, but I found ...
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0answers
31 views

Stuck with an optimization problem with 2 constraints (Lagrangian multiplier method)

I am really stuck with a certain minimization task. I thought I would understand the Lagrangian multiplier method (at least I could solve simple 2-variable optimization problems with 1 constraint). ...
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1answer
123 views

How to find the price elasticity of demand?

I need help answering if this is demand elastic of inelastic. A policy adviser suggests that in order to improve its balance of trade with china, Canada should lower the price of some heavy ...
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0answers
18 views

Vector Derivative of Dot Product in Reflection Equation

There are several questions on this network about differentiating a dot product of vector functions of an independent scalar parameter $t$:$$ \frac{d}{d t}\left(\vec{f}(t)\cdot\vec{g}(t)\right) = ...
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1answer
40 views

Finite difference numerical differentiation

I needed to find an O(h2) method to find f'''(x). Using Taylor expansions, I found: $$f'''(x)=\frac{f(x+2h)-2f(x+h)-2f(x-h)+f(x-2h))}{2h^3} + O(h^2)$$. However, I have also found that: ...
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1answer
43 views

Prove a sequence converges

Question: Let $f$ be differentiable on R with $a = \sup{|f′(x)| : x ∈ R} < 1$. (a) Select $s_0 \in \mathbb{R}$ and define $s_n =f(s_{n−1})$ for $n \geq 1$. Thus $s_1 = f (s_0 ), s_2 = f (s_1 )$, ...
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1answer
14 views

How to graph the derivative of this graph?

This is my professor's answers. I don't get the first dotted line on the derivative graph. There seems to be nothing wrong with the original graph in that place. I get the second dotted line because ...
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1answer
64 views

differentiability of $f(x,y)= \begin{cases} \frac{\sin^4(x)\sin^4(y)}{x^2+y^2} &\text{if $(x,y) \neq (0,0)$ }\\0&\text{if $ (x,y)=(0,0)$}\end{cases}$

$f:\mathbb{R}^2\rightarrow \mathbb{R}, (x,y)\mapsto \begin{cases} \frac{\sin^4(x)\sin^4(y)}{x^2+y^2} &\text{if $(x,y) \neq (0,0)$ }\\0&\text{if $ (x,y)=(0,0)$}\end{cases}$ (i) Show that $f$ ...
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0answers
164 views

Related Rates Word Problem (Boat+Rope)

A rope is attached to a boat at water level, and a person on the dock is pulling on the rope at a rate of $50 \text{ ft/min}$. If the person's hands are $16 \text{ ft}$ above the water level, how fast ...
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4answers
30 views

Derivative of a square root with exponential function

So I have the following function: $f(x)= \sqrt{e^{2x}}$ After applying the chain rule I sit with: $$\frac{1}{2\sqrt{e^{2x}}}2e^{2x}$$ From there I got: $$\frac{e^{2x}}{\sqrt{e^{2x}}}$$ While the ...
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2answers
84 views

Differentiating matrices with respect to a vector

Given a matrix $X$ (which doesn't need to be square) and a vector $b$, how can I get the following equality? $$\frac{b^t X^t X b}{\partial b} = (X^t X ) b $$ Why is this wrong? $$\frac{b^t X^t X ...
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2answers
54 views

Derivative of $\sin^2(x)$ first principles?

I know the first principle, $$f'(a) = \lim_{x\to a}\frac{f(x)-f(a)}{x-a}.$$ However, I don't know what to do next. Help.
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1answer
44 views

Finding derivative from (modified) Maclaurin series

I am given a function $$f(x) = \frac{x^{1521}}{x^2+7x+6}$$ I found the Maclaurin series to be $$x^{1521}\left(\sum_{n=0}^\infty \left(\frac15\left(-1\right)^n - ...
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2answers
19 views

$f(x)=x^9+3x^3+3x-3$, there is only on $c$ to $f(c)=2c$

Let $f(x)=x^9+3x^3+3x-3$. I want to show that there is only one $c\in(0,1)$ such that $f(c)=2c$. How can i prove this?
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2answers
26 views

Does the chain rule apply in inverse function derivatives?

My problem is finding the derivative of $y=\arctan (3x)$. Would it be $$y'= \dfrac{1}{1+(3x)^2}$$ or $$y'= \dfrac{1}{1+(3x)^2}\times 3$$
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1answer
147 views

Richardson Extrapolation - problems understanding how it works

I'm doing homework, and I am stumped on the first problem. I'm given this: Apply the extrapolation process described in Example 1 to determine $N_3(h)$, an approximation to $f(x_0)$, for the ...
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2answers
46 views

How to differentiate $y=\frac{2^x+4^x}{3^x+5^x}$

Differentiate $$y=\dfrac{2^x+4^x}{3^x+5^x}$$ I think you have to use implicit differentiation, but I don't know how to start. I first ln both sides and separated the fraction into ...
3
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0answers
63 views

Can this summation be expressed differently?

Lets say I have a sum that states the following $$ \sum_{j=0}^{k-c} {k-c \choose j}\ln(a)^{k-c-j} \frac{d^j}{dx^j}[(x)_c] $$ where $(x)_c$ is the falling factorial such that $$ (x)_c = ...
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1answer
75 views

Prove that $f^{-1}$ exists and is differentiable on $(0, ∞)$ for $f(x) = x^2e^{x^2}$.

Let $f(x) = x^2e^{x^2}$, and assume that $(e^x)' = e^x$ for all $x$ in $R$. a) Prove that $f^{-1}$ exists and is differentiable on $(0, ∞)$. Proof: Suppose that $f(x) = x^2e^{x^2}$, then finding ...
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1answer
77 views

Prove that $\frac{d^n}{dx^n} (\sin^4 x + \cos^4 x) = 4^{n-1}\cos (4x + \frac{n\pi}{2})$

Question Prove that $\frac{d^n}{dx^n} (\sin^4 x + \cos^4 x) = 4^{n-1}\cos (4x + \frac{n\pi}{2})$ My attempt First calculate $\frac{d}{dx} (\sin^4 x + \cos^4 x)$, that is, $$\frac{d}{dx} ...
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1answer
18 views

Help with simplifying implicit differentiation

Given the equation $\frac{y}{x+7y} = x^6 + 7$, find $\frac{dy}{dx}$. Ok, so I started to solve for $\frac{dy}{dx}$ and got to here: $\frac{\frac{dy}{dx}(x+7y)-(1+7\frac{dy}{dx})(y)}{(x+7y)^2} = ...
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3answers
24 views

Fairly simple differentiation question

Ok so the question is : If $f(x) = \frac {e^x} {x^6}$ Find $f'(x)$. I'm fine finding the answer, I know $\frac {e^x}{x^6} = e^x * \frac {1}{x^6}$ so I went ahead and used the product rule and got ...
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1answer
39 views

Maxima and minima of partial derivatives

I'm currently on the topic of finding maxima/minima for partial derivatives. However, I've recently come across a question which is rather confusing. Given: $$f(x,y) = x^3 -y^2 + 3x for (x,y) R^2 ...
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1answer
76 views

The values of $k$ for which $ \log(2x) \leq kx \leq e^{x/2}$ for all $x > 0 $

So I'm trying to solve a system of equations and I checked some other guys solution and he divides the function by the derivate, like so: $f(x)/f'(x)$. Find the values of the real constant $k$ for ...
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0answers
27 views

Total derivative of scalar function with respect to a vector

If i have a real valuad scalar function $f(y(x),z(x))$ with $y(x): \mathbb{R}^{n_x} \mapsto \mathbb{R}^{n_y}$ and $z(x): \mathbb{R}^{n_x} \mapsto \mathbb{R}^{n_z}$ and i want to get the total ...
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2answers
73 views

How to differentiate $y=x^{y^{\sin x}}$

I know I'll have to use implicit differentiation, but I always get stuck when there is an exponent with trig, log, and/or natural log.
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4answers
39 views

Let $z = x^a y^b \ln(xy)$. Find $x \frac {dz} {dx} - y \frac {dz} {dy}$ in terms of $z$

I'm baffled by this question. I assume I'm meant to use the product rule to work out $\frac{dz}{dx}$ and $\frac{dz}{dy}$? But when I'm doing that I'm getting crazy answers that I know are wrong: ...
3
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3answers
119 views

How to evaluate $\lim\limits_{x\to 0} \frac{\sin x - x + x^3/6}{x^3}$

I'm unsure as to how to evaluate: $$\lim\limits_{x\to 0} \frac{\sin x - x + \frac{x^3}{6}}{x^3}$$ The $\lim\limits_{x\to 0}$ of both the numerator and denominator equal $0$. Taking the derivative ...
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2answers
61 views

Concavity of function $F(x)=x^{1/5}(x+6)$ [closed]

Concavity of function $F(x)=x^{1/5}(x+6)$ The derivative's I have found don't seem to work. I found my zero value on my second derivative to be 4, but the program I am using says it is wrong. Help ...
3
votes
5answers
661 views

Chain rule with triple composition

We are supposed to apply the chain rule on the following function $f$: $$ f(x) = \sqrt{x+\sqrt{2x+\sqrt{3x}}} $$ I assumed we could rewrite this as $$ f(x) = g(h(j(x))) $$ However, I was not sure ...
2
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1answer
74 views

Analysis-Baby Rudin's differentiability and continuity: theorem 5.2 and 5.6

I am very confused about differentiability and continuity. At the beginning of the differentiation chapter, we proved that differentiability contains continuity. (Theorem 5.2) But in example 5.6 and ...
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1answer
83 views

Definition of Point of Inflection

An inflection point is a point on a curve at which the sign of the curvature (i.e. the concavity) changes. According to Wikipedia, "If x is an inflection point for f then the second derivative, ...
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2answers
33 views

Approximate $f(1.01)$ for a function satisfying $f'(x) = 3f(x) + 3x$ and $f(1)=3$

Suppose that the derivative of a function satisfies the formula $f'(x) = 3f(x) + 3x$. If $f(1)=3$, use linear approximation to estimate the value of the function at $1.01$. I think I found $f(x) = ...
0
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2answers
34 views

Linear Approximation.

Use linear approximation to approximate the number $ln(1.02)$. This is what I did and it is still wrong on my online homework. $f(x) = ln(x)$ $f'(x) = \dfrac{1}{x}$ $y=\dfrac{1}{x}(x-1)$ ...