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

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1answer
13 views

Partials of $f(t+\theta)$ evaluated at $t=0$ versus partials of $f(\theta)$

Suppose that $f:\mathbb{R}^n\to\mathbb{R}$ is infinitely differentiable with respect to its components. For $k\geq 1$ and $1\leq i_1,\ldots,i_k\leq n$, let's look at a $k$-th order mixed partial of a ...
1
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1answer
52 views

prove a polynomial is divisible

How do I prove that if a polynomial $p(x)$ is divisible by $(ax+b)^n$ where $n>1$ then $p'(x)$ is divisible by $(ax+b)^{n-1}$ I have no idea how to prove that but by logic it is obvious that is ...
-2
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2answers
50 views

Evaluate these limits by relating them to a derivative [closed]

Evaluate these limits by relating them to a derivative. $\lim\limits_{x \to 0} \frac{\sqrt{\cos{x}}-1}{x}$
1
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1answer
29 views

Continuity and Differentiability Q

We have f = e^(-1/|x|) if x is not equal to 0 and f(0) = p. Question 1: for what value(s) of p is f differentiable at 0? Question 2: is f' continuous for the values found in question 1? What I tried ...
1
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1answer
43 views

Can someone give me some guidance on this?

I have the function; $$ f(x) = \frac{x^2+1}{|x|+1}$$ I sketched the graph and saw that there are two local minima on it. They seem from the graph to be around $(x,y) = (-0.5, 0.825)$ and $(x,y) = ...
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4answers
149 views

How can I find $f'(2x)?$

it might seem a little bit elementary. $f$ is defined on $\Bbb R$ and it is differantiable. and is not equal to zero. if $xf(x)-yf(y)=(x-y)f(x+y)$ then find what is $f'(2x)$ equal to?. ...
1
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1answer
674 views

Related Rates - Particle Moving along a parabola

A bug is moving along the right side of the parabola y=x^2 at a rate such that its distance from the origin is increasing 3 cm/min. At what rate are the x- and y-coordinates of the bug increasing when ...
2
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2answers
35 views

Proof that $f'(0)$ of an even function is always $0$.

$f \in C^1$ and "even" meaning that $\forall x:f(x)=f(-x)$. To me it seems logical, but I'm struggling with writing it down. Does this work? $$f'(0)=\lim_{x\rightarrow 0}f'(x)=\lim_{x\rightarrow ...
1
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0answers
33 views

Analytical expression for the value where maxima of the function occurs

I'm trying to find the expression of $\theta$ for which the following equation is maximized. $$f(\theta)=\{a-b~cos\theta-c~sin\theta\}^{-\eta}+\{a-b~cos\theta+c~sin\theta\}^{-\eta}$$ Here, ...
0
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0answers
28 views

The derivative of the dual of the linear regression with respect to alpha

So for the optimization min ξ,w $\sum_{i=1}^{n} {ξ_i}^2 + λw^Tw$ subject to: $w > x_i − y_i − ξ_i = 0, i = 1, ..., n$ I derived the dual of this optimization to be $-\lambda^2 \sum_{i} \beta_i^2 ...
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0answers
31 views

Differentiable functions as direct summand of continuous ones

When I use notion space, I mean just vector space over $\mathbb{R}.$ So I have a point on real line (lets say it is $0$) and space of continuous funtions around that point. I consider subspace $I$ of ...
0
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1answer
24 views

Differentiating $\sqrt{x}+\sqrt{x^3}+\sqrt[3]{x^2}$

I'm new to derivatives and I'm already having trouble with the exercises: $$\sqrt{x}+\sqrt{x^3}+\sqrt[3]{x^2}$$ So, given that it's a sum of functions, I can split this in three parts, yes? For ...
2
votes
2answers
35 views

Instantaneous rates of change

I am having problems solving the following question. The volume, $V$, of a sphere of radius r is given by $V=f(r)=\frac{4}{3}\pi r^3$. Calculate the instantaneous rate of change of the volume, $V$, ...
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0answers
18 views

Notation regarding the interior product

Would someone be so kind as to explain how one would obtain the right hand side and perhaps elaborate on the notation: $v\lrcorner d(\xi_i dx^i)+d(v\lrcorner \xi_i dx^i)=v^j\xi_{i,j} dx^i -v^i ...
2
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3answers
91 views

Finding abs max with trig function

I have $g(x)=\sqrt{3}-2\cos(x)$ on $[0,\pi]$. I did $g'(x)=-(-2\sin(x))=2\sin(x)$ and then $\pi$ and $0$ as the critical numbers. I evaluated the original function at each of the critical numbers ...
1
vote
1answer
43 views

Why is the derivative used to represent marginal cost instead of the difference?

Marginal cost is informally defined as "the change in the total cost that arises when the quantity produced is incremented by one unit." And given a total cost function $C(q)$ that's differentiable, ...
0
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1answer
48 views

Calculate a cubic curve from two slopes and two points

I have a question that is supposed to be solved using derivatives. The question asks to find the cubic polynomial function $f(x)$ where $y=13(x-1)+4$ is tangent to $f(x)$ at $(1,4)$ and $y=(x+1)+6$ is ...
7
votes
2answers
223 views

Why is it wrong to derive the chain rule this way?

My book says that the chain rule can stated as $$\dfrac{dy}{dx} = \dfrac{dy}{dt} \dfrac{dt}{dx}$$ However, it the book says that it is incorrect to reason that the chain rule is true because the ...
0
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1answer
32 views

Solving a pair of kinetic equations using the Laplace transform

I have the following set of kinetic equations: \begin{align} \frac{dx}{dt}&=r_1\delta(t-t_0)-(r_2+r_3)x(t)\\ \frac{dy}{dt}&=r_3x(t)-r_5y(t). \end{align} How can I solve for $y(t-t_0)$ using ...
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0answers
20 views

Fixed points of successive differentiation.

Let $f(x)$ be a function and $$\mathfrak{F}:C^{\infty}\rightarrow C^{\infty} : f(x)\mapsto\lim_{n\rightarrow\infty}\frac{d^nf}{dx^n}$$ when it exists. Then what is the domain of $\mathfrak{F}$? I ...
1
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1answer
42 views

Error in logarithmic differentiation of $R(s)=s^{\ln s}$

I was trying to solve for the derivative of $R(s) = S^{ln(s)}$. I understand that there is a much simpler way to do it through a single use of the chain rule, but I wanted to see if I could figure out ...
4
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1answer
49 views

Derivative of power series with nonnegative coefficients

Let $$f(x) = \sum_{k=0}^\infty a_kx^k$$ be a power series mapping reals to reals, with radius of convergence $1$. Suppose $f'(x_0)$ exists in $(-1,1]$ (take the one-sided limit if $x_0 = 1$). Also ...
2
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1answer
41 views

The derivative of a two-to-one complex function has no zeros.

Let $U$ be open in $\mathbb{C}$ and $f \in H(U)$ such that $f$ is two-to-one on $U$. Prove that $f'$ has no zeros in $U$. I am thinking about using Cauchy's Integral formula for derivatives, but I ...
0
votes
2answers
44 views

How to take the derivative of $Y=\log(x+\sqrt{a^2+x^2})$?

$$Y=\log(x+\sqrt{a^2+x^2})$$ Find $\dfrac {dY}{dx}$. My answer: $$\dfrac 1{x+\sqrt{a^2+x^2}}\cdot\frac{d}{dx}\sqrt{a^2+x^2})$$ Which again goes to a chain rule as $\frac{2}{3} ...
0
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1answer
36 views

Partial derivative of composition with multivariable function.

I think this should be an easy question? However, a quick google search has not revealed the confirmation I want, so I am posting here. If we have a function $f(x,y)$ and a monotonic transformation ...
1
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1answer
34 views

What does the second value of `x` mean here?

I looked into this fun math question and I understood his whole explanation except for one part. When he sets $T'(x)=0$, he eventually concludes to $x=8$ and $x=0$. What does the $x=0$ mean? I ...
2
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1answer
67 views

Derivative of power series

Let $$f(x) = \sum_{k=0}^\infty a_kx^k$$ be a power series mapping reals to reals, with radius of convergence $R$. Suppose $f'(x_0)$ exists in $[-R,R]$ (take the one-sided limit if $x_0 = R$ or $x_0 = ...
0
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1answer
34 views

Using the chain rule of differentiation to evaluate an integral along a curve

I have a little confusion regarding the following: $\gamma $ is a piecewise smooth curve from $A$ to $B$ and $h(x,y)$ is a continuously differentiable function on $\gamma$. Let ...
0
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0answers
50 views

Factorial moment and derivative of probability generating function

The $k$-th factorial moment of a non-negative integer random variable $X$ is given by the expectation $E[(X)_k] = E[X(X-1)\cdots(X-k+1)]$. The probability generating function of the same $X$ is given ...
0
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0answers
30 views

Function satisfing : $h(x)=f(2x-1)$ with $f'(-1)=0 $ and $f'(2)=-2$ then what is $h'(x) $?

I find in some book this function defined as follow $h(x)=f(2x-1)$ . with $f'(2)=-2 $ and $f'(-1)=0$ , we would like to know $h'(3/2)$ ? In the book take $h'(x)=f'(2x-1)=(f'(2))x+f'(-1)=-2x $ but ...
1
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1answer
21 views

Help in finding error in derivative quotient rule

So, I have $f(x)=u(x)v(x) \implies f'(x)=u(x)v'(x)+v(x)u'(x)$, and also have $f(x)=\frac{1}{x} \implies f'(x)=-\frac{1}{x^2}$. I cannot see what is so then wrong with $f(x)=\frac{u(x)}{v(x)} ...
2
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2answers
32 views

A version of the product rule

Using the product rule we know that $$\frac{ {\rm d}\ln(fg)}{ {\rm d} x} = \frac{f'g+fg'}{fg}$$ Is there a function $K$ such that $$\frac{ {\rm d} K(f,g)}{ {\rm d} x} = \frac{f'g-fg'}{fg}$$ ...?
4
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0answers
40 views

Linear algebra machinery for differentiation of families of functions.

So I know that since differentiation is linear, for many types of functions we can represent it using linear algebra. Famous examples include polynomials, if we represent them with their coefficients ...
7
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2answers
191 views

$f'\in \mathcal R([0,1])$ , then $\lim_{n \to \infty} \sum_{k=1}^n f\Big(\dfrac kn \Big) - n \int_{0 }^1 f(x)dx=\dfrac{f(1)-f(0)}2$?

If $f:[0,1]\to \mathbb R$ is a differentiable function with continuous derivative then I can show that $$ \lim_{n \to \infty} \left[ \sum_{k=1}^n f\!\left(\dfrac kn \right) - n \int_0^1 f(x)\,dx ...
0
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1answer
29 views

How to complete this partial differential equation?

$T = \frac{1}{2}M_{w}\dot{x}^{2} + \frac{1}{2}I_{w}\frac{\dot{x}^2}{r^2} + \frac{1}{2}M_{b}((\dot{x} + l\dot{\theta}cos(\theta))^2 + (l\dot{\theta}sin(\theta))^2) + \frac{1}{2}I_{b}\dot{\theta}^{2}$ ...
0
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6answers
62 views

How do I find the derivative of $\frac{1}{2\sqrt{x}}$ using the definition of the derivative?

Using the definition of the derivative and not the power rule, how would I find $f(x) = \frac{1}{2\sqrt{x}}$
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2answers
42 views

Why isn't $df=\frac{\partial f}{\partial x}\:dx+\frac{\partial f}{\partial y}\:dy$ defined to resemble a Taylor series further?

I'm not sure if this is a duplicate (it might even just be silly), but why isn't the differential of some function $f\left(x_1,x_2\right)$, \begin{align}df&=\frac{\partial f}{\partial ...
0
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1answer
45 views

Breaking a biexponential function in two

I have the following two equations: $$ I(t) = \sum_iw_i\alpha(t-t_i) $$ $$ \alpha(t) = \beta\frac{\tau_2}{\tau_2-\tau_1}(e^{-t/\tau_1}-e^{-t/\tau_2}). $$ When implemented in a particular software ...
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1answer
28 views

Give a formula that shows that $u(x,y)=ax^2+bxy+cy^2$ is the real part of a function of the form $f(z)=Az^2$

Show that $u(x,y)=ax^2+bxy+cy^2$ is the real part of a function of the form $f(z)=Az^2$ for some $A \in\mathbb C$. Give a formula for $A$ in terms of $a, b$ and $c$. I first needed to prove that $u$ ...
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3answers
88 views

Derivative Notation Explanation

I am learning differential calculus on Khan Academy, but I am uncertain of a few things. (by the way, I understand derivatives this far: $d'(x)$ and this: $d'(g(x))$) I am confused mainly about ...
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6answers
88 views

How do I use the power rule with $4x^2 - 2$?

Every website I look at only shows the power use with a single term example or polynomial. Could somebody please give me a step by step breakdown of using the power rule to find $f'(x)$ where $f(x) = ...
1
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1answer
35 views

Just one Help needed [closed]

ArcSin{(1-x^ 2)/(1+x^2)} To find derivative what should we put for x? Is 1-cos^2(2x)=sin^2(2x)?
4
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3answers
62 views

If $\sqrt{1-x^2}+\sqrt{1-y^2}=a(x-y)$, then what is $\frac{dy}{dx}$?

The question states: If $\sqrt{1-x^2}+\sqrt{1-y^2}=a(x-y)$, then what is $\frac{dy}{dx}$? The options are: $$\sqrt{\frac{1-x^2}{1-y^2}}$$ $$\sqrt{\frac{1-y^2}{1-x^2}}$$ ...
3
votes
2answers
40 views

Derivative of the inverse of a matrix

I've seen in a scientific paper this equation: $\frac{\delta K^{-1}}{\delta p} = -K^{-1}\frac{\delta K}{\delta p}K^{-1}$ where K is a $n\times n$ matrix which depends on $p$. In my calculations I ...
1
vote
1answer
43 views

linear approximation of $f(x)$

Let $y=f(x)=(x_1^2+2x_2, x_1x_2-3x_1)$ Is the linear approximation just $f(y)=f(x)+A(y-x)$ whenever $y$ is approximately near $x$? I know that if I calculate the Jacobian matrix, I can get that ...
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2answers
20 views

Estimating values of a function by considering the graph of its derivative

Let the function $f$ be continuous in $[0,4]$ and differentiable in $(0,4)$. Looking at the graph I am so tempted to say that $f(2)=f(4)$ but I guess $f(2)>f(4)$ is correct. I know this is very ...
1
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1answer
19 views

checking differentiability on a multivariate function

At what points would $f$ be differentiable? $f(x_1,x_2)=\begin{cases} \dfrac{x_1^3}{|x_1|+|x_2|} \text{ for } (x_1,x_2)\neq (0,0)\\ 0\text { if }(x_1,x_2)=(0,0) \end{cases}$ Only place that I can ...
0
votes
1answer
31 views

The integral of $\frac{1}{\tau_1-\tau_2}(e^{-t/\tau_1}-e^{-t/\tau_2})$

If we have $$ \frac{dx}{dt}=\frac{1}{\tau_1-\tau_2}(e^{-t/\tau_1}-e^{-t/\tau_2}), $$ what is $x$, and what are the steps by which one comes to that solution?
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vote
5answers
75 views

The derivative of $1 - e^{-t/\tau}$

I am failing to understand how to compute the derivative of a few exponential functions. Let's start with this one: $$ v = 1 - e^{-t/\tau} $$ The derivative is $$ \frac{dv}{dt} = \frac{1-v}{\tau} ...
0
votes
0answers
27 views

Find the critical point of a 2 variable function

I have this function: $f(x,y)=4x^3+2xy^2-x^2+4y^2$ The partial derivative of x and y are: $\dfrac{\partial f}{\partial x}=12x^2+2y^2-2x$ $\dfrac{\partial f}{\partial y}=4xy+8y$ They ask me to ...