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

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2answers
18 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?
1
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2answers
23 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 $$y'= \dfrac{1}{1+(3x)^2}\times 3$$
1
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1answer
58 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 ...
0
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2answers
41 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
votes
0answers
56 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 = ...
1
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1answer
41 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 ...
3
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1answer
69 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} ...
1
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1answer
14 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} = ...
0
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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 ...
2
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0answers
28 views

How to apply the chain rule in this case? Infinite derivatives?? [closed]

I have the "fucntional" $$ P = P\left\lbrace M\left[\bar{\mu}(\mu, M)\right]\right\rbrace, $$ I want to calculate the total derivative $dP/d\mu$. $M$ and $\bar{\mu}$ are a interrelated self ...
0
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1answer
29 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 ...
0
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1answer
68 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 ...
0
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0answers
10 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 ...
1
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2answers
72 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
34 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
votes
3answers
113 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
54 views

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

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 ...
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1answer
31 views

differentiation [closed]

it is given that the dimensions of a parallelepiped at a certain instant are X=4cm , Y=3cm and Z=12cm if at this instant X increases at rate 1cm/sec , Y increases at rate 0.5cm/sec and Z decreases at ...
3
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5answers
282 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
53 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 ...
1
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1answer
26 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
31 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) = ...
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2answers
25 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. $f(x) = ln(x)$ $f'(x) = \dfrac{1}{x}$ $y=\dfrac{1}{x}(x-1)$ ...
4
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1answer
57 views

Prove that if $f$ is differentiable at $x=0$, then $f$ is differentiable on $\mathbb{R}$.

$Conj:$ Suppose that a function $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable at $x=0$, satisfies $f(a + b) = f(a)f(b)$ for all $a,b,\in\mathbb{R}$, and is not identically zero ($\exists ~x$ ...
4
votes
1answer
180 views

The proper and easiest way of doing an integral with derivative?

I have this integral: $$\int{\sec^3x\,\mathrm dx}$$ I don't understand how I would solve this. Google and YouTube videos don't help me understand much, other than just giving the answer. Is it ...
3
votes
1answer
35 views

How do I Implicitly Differentiate this equation?

My equation is $y=x^{y^2}$ I did the $\ln$ of both sides, then I tried implicit differentiation. I got $$y'= \frac{x^{y^2} y^2}{x}.$$
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0answers
17 views

Multivariable chain rule with vector valued function

Suppose $f:\mathbb{R}^n \rightarrow \mathbb{R}$, $\mathbf{g}:\mathbb{R}^n \rightarrow \mathbb{R}^n$ and $\mathbf{x} \in \mathbb{R}^n$. How do I find a formula for $\nabla f(\mathbf{g}(\mathbf{x}))$?
1
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1answer
21 views

Differentiate this equation (below):

$$\Large y = x^{\ln 7} + \log_7 x $$ I know for differentiating logarithms you do: $1/f(x) \cdot f'(x) \cdot 1/\ln b$. But how about differentiating $x^{\ln 7}$? I don't understand how to change ...
6
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3answers
66 views

Solve for constants: Derivatives using first principles

Question Find the values of the constants $a$ and $b$ such that $$\lim_{x \to 0}\frac{\sqrt[3]{ax + b}-2}{x} = \frac{5}{12}$$ My approach Using the definition of the derivative, $$f'(x) = ...
1
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1answer
30 views

evaluation of $\nabla \cdot ( \boldsymbol{B}(\boldsymbol{x}) \cdot \boldsymbol{B}^{T}(\boldsymbol{x}) )$

i have a symmetric positive matrix $\boldsymbol{D}(\boldsymbol{x})$ which can be decomposed as: $\boldsymbol{D}(\boldsymbol{x})$ = $\boldsymbol{B}(\boldsymbol{x}) ...
0
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1answer
29 views

Constructing Manifolds: Submersion

Given a smooth manifold $M$ and a topological space $N$. Consider a local homeomorphism $F:M\to N$ with $\mathrm{im} F=N$. Then one can turn the target space into a smooth manifold via: ...
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0answers
24 views

significance of L1 norm of gradient vector

I was wondering if anyone here knew any mathematical paper which 1) uses properties or uses the L1 norm of the gradient (of any function) vector? 2)there any quantity defined in literature that ...
1
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2answers
63 views

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

I was wondering when this function would curve upwards/downwards. I was having trouble finding the inflection points. Thank you. $$F(x) = x^{1/5} (x+6)$$ Progress I found the first derivative to be ...
0
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0answers
25 views

Expansion of Matrix power

A,B are skew symmetric matrices with determinant $0$ $$A=\left( \begin{array}{ccc} 0 & -c_0 & b_0 \\ c_0 & 0 & -a_0 \\ -b_0 & a_0 & 0 \\ \end{array} \right).$$ ...
1
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1answer
26 views

Functions of several variables and $Df$

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a smooth function and let $g:\mathbb{R}^n \rightarrow \mathbb{R}$ be defined by $g(x_1,...,x_n)=x_1^5+...+x_n^5$. Suppose $g\circ f\equiv 0$. Show that ...
1
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2answers
46 views

Use limit definition to find derivative of $x+\sqrt x$

The function is $f(x) = x + \sqrt x$. How would you use the limit definition of the derivative to find the derivative of that equation?
1
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1answer
28 views

Divergence of a vector field?

For some vector field $F = f(x)i + g(y)j$, the divergence in $\mathbb{R}^2$ is defined by: $\frac{\partial {f}}{\partial {x}} + \frac{\partial {g}}{\partial {y}}$. What happens if $f$ or $g$ is not ...
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6answers
60 views

If $f(x)\ge g(x)$, is $f'(x)\ge g'(x)$?

We choose any function for $f(x)$ and $g(x)$. Also, $x$ needs to be positive at all times. Lets say that $f(x)=45x^2$ and $g(x)=15x^2$. We can say that $f(x)\ge g(x)$, if $x\ge 0$. So the condition ...
16
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2answers
778 views

A functional equation with no solution

Let $f:\mathbb{R}\to (0,\infty)$ be a differentiable function satisfying $$f(f(x))=f^\prime(x)$$for each $x$. Show no such function exists. I got this problem in an exam. I haven't done anything ...
0
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1answer
23 views

Stream functions and divergence?

We see that the existence of a stream function guarantees that the vector field has zero divergence or, equivalently, is source free. The converse is also true on simply connected regions of ...
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2answers
50 views

Solving $x\sin(\frac 1x)$ via limit definition

I'm trying to show that the derivative of $x\sin(\frac 1x)$ exists and is equal to $\sin(\frac 1x)-\frac {\cos(\frac 1x)}x$ for every point in its domain via the limit definition (I can of course just ...
0
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0answers
15 views

Computing Jacobian of error function using Lie Algebra

First off all, I hope this is the right place to ask, as it is a computer vision problem, but I'm specifically asking about the mathematical part of it. I am currently implementing the ICP (Iterative ...
4
votes
2answers
100 views

Computers can't deal with limit of $\Delta x \to 0$

While I was studying about finite differences I came across an article that says "computers can't deal with limit of $\Delta x \to 0$ " in finite differences.But if computers can't deal with these ...
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0answers
31 views

Critical Points of a Complex Sine Function within Bounds

I need a method to find the critical points of the function below. f(x) = 3.8*sin(2.4*x + 1) - 2.3*sin(7.2*x - 2) + 3.2*sin(8.1*x - 3) Bounds [-10, 10] I ...
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1answer
16 views

Complex Analysis - Complex plane, differentiable

Determine all the points in the complex plane where the function f(z) = tan(z) is differentiable and calculate the derivative at those points.
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0answers
14 views

Extreme-Value Theorem?

The extreme-value theorem states that if a function $f(x_{1},x_{2},...x_{n})$ is continuous in a closed and bounded interval within the domain of $f$, there exists both an absolute maximum and ...
0
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3answers
32 views

How can I determine a general formula for the nth derivative of any continuous function f(x) differentiable at least n times?

I know how to do it with easier functions, but is there a universal method which can be applied to all continuous functions differentiable at least n times(introduced to in a second year calculus ...
2
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1answer
40 views

Inflection point not found for the function $f(x) = 2\arctan(x) - \dfrac{x^3}{x^2+1}$. Should it?

$f'(x) = -\dfrac{x^4+x^2-2}{\left(x^2+1\right)^2} = \dfrac{(x+1)(x-1)(-x^2-2)}{\left(x^2+1\right)^2}$ This gives the critical points $x=-1 \quad\&\quad x=1$. Solving those with sign analysis; ...
2
votes
1answer
29 views

how to maximize weekly revenue using profit function and derivatives

p=45-0.01q where p is price of each product sold and q is the quantity of products sold. a) find the quantity that maximizes the weekly revenue of the company b) what price should the company sell ...
4
votes
1answer
49 views

Solve $f'(x) = 0$ and set up a sign chart for $f'$.

I understand how my teacher got the two $x$ values, but why didn't he solve for $e^x=0$? I know he did $x=0$ which is $0$ $x+2=0$ which is $-2$ so why no $e^x=0$? is there even an answer for ...