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

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2answers
37 views

First Derivative of a Summation

$\frac{k}{n}\sum_{k}^{n-1}\frac{1}{i}$ What is the first derivative of this with respect to k? Thank you
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1answer
34 views

derivative power rule

I'm looking at a programming text that says the following: The rule for differentiating powers says that the derivative of $[u(x)]^n$ with respect to $x$ is equal to $n[u(x)]^{n-1}$ times the ...
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2answers
60 views

Calculus Derivatives Problem [closed]

Can anyone find infinitely many pairs of functions f(x), g(x) such that (f(x)*g(x))' = f'(x)*g'(x) Also, neither f'(x) nor g'(x) can equal 0. (They can't be constants) Edit by Igor Minevich: This ...
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4answers
51 views

How do I use implicit differentiation?

I have to find the derivative of $y=\cos^{-1}(11 x^{12})$. Should I use implicit differentiation with respect to x here, or is there another way to go about this?
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1answer
32 views

Tangent at point $P_1$ other than $(0,0)$ on the curve $y =x^3$ meets the curve again at $P_2$…

Problem : Tangent at point $P_1$ other than $(0,0)$ on the curve $y =x^3$ meets the curve again at $P_2$ The tangent at $P_2$ meets the curve again at $P_3$ and so on. If $\frac{\text{area} ...
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1answer
26 views

Find $g'(\frac{-1}{2})$ and $g''(\frac{-1}{2})$

Let $f(x)=\frac{x^3}{x^2+1}$, and $g(x)$ is the inverse function of $f(x)$. Then $f(-1)=\frac{-1}{2}$ and $g(\frac{-1}{2})=-1$. Find $g'(\frac{-1}{2})$ and $g''(\frac{-1}{2})$. I have found ...
3
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1answer
39 views

Prove , there exists $\theta , \phi \in (\frac{\pi}{6},\frac{\pi}{3})$ such that $f'(\theta) = 0$ and $f'(\phi)\neq 0$

Let the function $$f(\theta) = \begin{vmatrix} \sin\theta & \cos\theta & \tan\theta \\ \sin(\frac{\pi}{6}) & \cos(\frac{\pi}{6}) & \tan(\frac{\pi}{6}) & \\ \sin(\frac{\pi}{3}) ...
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1answer
27 views

Is the remainder of first-order Taylor expansion still continuously differentiable?

Let $f: {\mathbb R}^n \to {\mathbb R}^n$ be a continuously differentiable function. Then, we can rewrite its first-order Taylor expansion at $x \in {\mathbb R}^n$ for $h \in {\mathbb R}^n$ that ...
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1answer
21 views

Multivariable derivatives of two functions

Problem 2. Let the pressure $p$ and temperature $T$ at a point $(x,y,z)$ be $$P(x,y,z)=\frac{x^2+2y^2}{1+z^2},\quad T(x,y,z)=5+xy-z^2$$ a. If the position of an airplane at time $t$ is ...
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1answer
19 views

How do we find the inflection points of g?

Consider the function $g(x)=3x(x-2)^{\frac{2}{3}}$ How do we find all the inflection points? I found $$g^{\prime\prime}(x)=\frac{2(5x-12)}{3(x-2)^{\frac{4}{3}}}$$ so when I do ...
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2answers
34 views

Finding a differential equation

How do I find a differential equation for this equation: $ax+(y-b)^2=0$ I've tried deriving for $x$ and $y$ but it didn't work out very well.
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10answers
777 views

What is the use of Calculus? [closed]

I know this may seem like a really broad question, but I will narrow it down. I really want to know the purpose of some of the things my teacher is emphasizing in my calc class. For example why it ...
2
votes
3answers
55 views

Definition of limit with$ f(x)=|x^3|$

Using the definition of the limit I tried to find the derivative of $f(x)=|x^3|$. I came up with: $$f'(x)=\frac{3x^5}{|x^3|}$$ Question: Why is the derivative (according to this answer) not defined ...
7
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5answers
183 views

How to get nth derivative of $e^{x^2/2}$

I want to calculate the nth derivative of $e^{x^2/2}$. It is as follow: $$ \frac{d}{dx} e^{x^2/2} = x e^{x^2/2} = P_1(x) e^{x^2/2} $$ $$ \frac{d^n}{dx^n} e^{x^2/2} = \frac{d}{dx} (P_{n-1}(x) ...
1
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3answers
31 views

For f(x) = ((3x+7)^8)((4x-5)^3) , find f'(x) and use this answer to find the value(s) of x at which the graph of f(x) has a horizontal tangent line

I know that $f'(x) = (24(3+7)^7)*(4x-5)^3+((3x+7)^8)*12(4x-5)^2$, but is there any easier way to find the horizontal tangent line without expanding the terms using pascal's triangle and solving for x ...
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4answers
75 views

What is the derivative of $x^y$ with respect to $y$?

I'm looking for the derivative of $x^y$ with respect to $y$. I have done it by taking log of both sides, how do I do it if I try to write $\log e^{x^y} = x^y$?
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1answer
30 views

Find possible values of a and b with derivates.

The question is: Given that $y = ax^2 + bx$ and $\frac{d^2y}{dx^2} = 4(\frac{dy}{dx})^2 - 32y$ Find possible values for constants a and b. I worked out the first derivative to be 2ax +b and the ...
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0answers
64 views

Pointwise convergence of derivatives

Let $f_n$ be a sequence of functions converging pointwise to a function $f$ (all defined over a fixed interval $I$, eg $[0,1]$). Assume each $f_n$ is differentiable, and that $f_n'$ also converge ...
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1answer
33 views

Hyperbolic PDE that is Non-Homogenous

Say I have a PDE of form $$Au_{xx} + Bu_{xy} + Cu_{yy} = xy$$ and that A,B, and C are constants chosen so that the PDE is hyperbolic. How do I go about solving this system? I solved the homogeneous ...
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1answer
147 views

Find a strictly increasing function $ f$ with $ f'(1)=0$

Find a strictly increasing function $ f$ with $ f'(1)=0$. I've found the function $f(x)=\frac{x^3}{3}−x^2+x$ But I don't know how to prove that the function is strictly increasing.
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1answer
69 views

How to deduce the recursive derivative formula of B-spline basis?

Description Let $\vec{U}=\{u_0,u_1,\ldots,u_m\}$ denotes a non-decreasing sequence of real numbers, i.e, $u_i\leq u_{i+1} \quad i=0,1,2\ldots m-1$. and the $i$-th B-spline basis function of ...
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1answer
50 views

Why in a directional derivative it has to be a unit vector

Could you please explain me why when we compute the directional derivative we have to use the unit vector (u). I know that using 2u would change the directional derivative as the second point is ...
5
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0answers
63 views

How is $\frac{ds}{dt}$ related to $\frac{dx}{dt}$?

The problem states: Let $x$ and $y$ be differentiable functions of $t$, and let $s = \sqrt{4x^2+6y^2}$ be a function of $x$ and $y$. How is $\frac{ds}{dt}$ related to $\frac{dx}{dt}$ if $y$ is ...
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1answer
12 views

Solving $\frac{dV}{dt} = \frac{I}{C}$ using the Laplace transform

I have the following equation for the evolution of the membrane potential ($V$) of a neuron: $$ \frac{dV}{dt} = [-g_L(V-V_{rest}) + I_{syn}(t) + I_0] / C. $$ According to Equation 2.13 of this ...
0
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1answer
53 views

The total derivative of a complex function

Let $U$ be an open subset of $\mathbb{C}$, $a \in U$ and the function $f: \mathbb{R} \to \mathbb{C}$ is complex differentiable in $a$. a) Show that for some $u,v \in \mathbb{R}$ $Df(a)= ...
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4answers
53 views

How do you find the equation of a tangent line to an equation: $\sin$ in it?

This is the question I need to answer, but I don't know how to. Find an equation of the tangent line to $y=10\sin(x)$ at $x=\pi$.
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2answers
40 views

Understanding how to set up g(f(x)) comparatively to f(g(x))

The question reads: Given the following functions: $f(x)=\cos(x)$ and $g(x)=x^{7}+1$, find: a: $\displaystyle \frac{d}{dx} f(g(x)) = ?$ b: $\displaystyle \frac{d}{dx} g(f(x)) = ?$ ...
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2answers
28 views

Explaining the chain rule when using quotient rule.

I have a problem in my WebWorks hw , modeled after Rogawski ET 3e section 3.7, exercise 35. The problem reads, find the derivative of $(\frac{x+81}{x-81})^{20}$. I know the answer is $-3240 ...
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2answers
39 views

How does one prove differentiability?

If I have a piecewise function, must I prove it is continous to show it is differentiable at a point? Or is it if I am able to apply the derivative rules to the function, it must be continous and ...
2
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3answers
29 views

Calculate the value of a derivative at the origin

I have the following question in a course: An example of the logistic function is defined by $$\varphi(v)=\frac{1}{1+e^{-av}}$$ whose limiting values are $0$ and $1$. Show that the derivative of ...
5
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2answers
150 views

Second derivative of $\int_\mathbb{R}\cos(tx)dp(x)$

Let $p$ be a probability on $\mathbb{R}$ and $$f(t):=\int_\mathbb{R}\cos(tx)dp(x).$$ I want to show that if $f''(0)$ exists then $$f''(0)=\lim_{t\to 0}2\frac{f(t)-1}{t^2} \: \:(\star).$$ By ...
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1answer
22 views

What's the usual convention for directional derivatives in the $0$ direction?

Depending on the source, the definition of the directional derivative does not include the restriction that the direction vector be of unit length. In this case, it seems to me that we can then in ...
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3answers
42 views

Graph and derivative of y=x

I'm having trouble understanding why the graph of $y=x$ is different from the graph of $y = \sqrt{ x^2 }$. Aren't both equations the same once you simplify the second one? And isn't the derivative of ...
1
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1answer
44 views

Lyapunov invariant set for affine systems

Given a linear system $\dot{x}=Ax$ such that the real part of every eigenvalue of $A$ is less than $0$, Lyapunov's equation $A^T P + P A = -Q$ with $Q$ being any suitably sized positive definite ...
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1answer
10 views

Am I correct in writing the chain rule for univariate functions this way?

I was just making some notes on an online course for myself, and (trying to remember my university calculus), wrote down the chain rule this way: $$ \frac {\textrm{d}}{\textrm{d}x}f(g(x)) = \frac ...
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1answer
15 views

Deriving Relativistic Force

Newton’s law states that $F=\frac{dp}{dt}$, where $p$ is the momentum of a body. In Newtonian physics, if the body has constant mass $m$, its momentum is $mv$, and Newton’s law becomes the familiar ...
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1answer
69 views

Doppler Effect Related Rates Application

A police car is sounding a siren with a frequency of $1280$ Hz while traveling right towards you. At a certain time, you measure the frequency of the siren to be $1400$ Hz, and increasing at a rate ...
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2answers
64 views

Show that $f(x,y)$ is differentiable at $(0,0)$

Show that $f(x,y)$ defined by: $$f(x,y) = \begin{cases}\dfrac{x^2y^2}{\sqrt{x^2+y^2}}&\text{ if }(x,y)\not =(0,0)\\0 &\text{ if }(x,y)=(0,0)\end{cases}$$ is differentiable at $(x,y) = (0,0)$ ...
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1answer
45 views

Arc Length Derivatives

I've got a couple of questions regarding derivatives and the arc length formula. I've been given the arc length formula (where $s$ equals the integral from $x$ to $1$ of $\sqrt{1+(dy/dt)^2}dt$) I've ...
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3answers
74 views

How do you find the derivative of the integral $\sin(\ln x)$

$$\frac{d}{dx} \;\left[ \int_{a}^{x^2}\sin(\ln(z))\;dz\right]$$ I'm not sure if I'd have to do the chain rule on the natural logarithm and them $x^2$, or if there is no chain rule at all. Any help ...
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1answer
50 views

find the derivative of (cosx)^(sinx)^x

Solve to find the derivative of the following function: (cosx)(sinx)x Do not simplify the answer.
2
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1answer
30 views

$2^\text{nd}$ Derivative of normal distribution, evaluated at one standard deviation

What is the $2^{nd}$ derivative of the normal distribution at one standard deviation? The normal distribution is given by $N(x,\mu ,\sigma)=\frac{1}{\sigma\sqrt{2\pi ...
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0answers
29 views

How can I differentiate this intergal?

I am trying to replicate the result of an economics paper called Endogenous Technical Change from Paul Romer. In page 15 the author maximizes with respect to $x(i)$ the following: $max_{x(i)} ...
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0answers
45 views

Hessian of function

I need to calculate the Hessian of the following function: $f(x) = \ln\sum\limits_{a \in A}e^{\langle x,a\rangle}$ where $ x \in \mathbb R^n $ and $A$ a set of $n$ dimensional vectors. I calculated ...
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1answer
26 views

Antiderivative of fraction, wrong coefficient

$$\frac{x^2-\sqrt{x}-e}{\sqrt{x}}$$ Is what I need the antiderivative of. I separated it into $$\frac{x^2}{x^{1/2}}-\frac{x^{1/2}}{x^{1/2}}-\frac{e}{x^{1/2}}$$ then to $$x^{1/2}-1-ex^{-1/2}$$ and ...
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0answers
15 views

Partial and total derivative in an elasticity

I'm aware of the difference between partial and full derivatives. However, when I need to show that the following holds I fail. (This example comes form intertemporal substitution in macroeconomics.) ...
1
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5answers
149 views

Find the cubic function whose graph has horizontal Tangents

Problem: Find the cubic function $y = ax^3 + bx^2 + cx + d$ whose graph has horizontal tangents at $(-2, 6)$ and $(2, 0)$. Now I can never seem to gather enough information for find all the values of ...
0
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1answer
28 views

Finding the optimal value of g by differentiation

I'm currently reading the book "Introduction to Information Retrieval" (http://nlp.stanford.edu/IR-book/). Chapter 6.1 is about finding the optimal weight g for a specific function. The function given ...
2
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1answer
54 views

Proof: Need help with Rodrigues's formula for finding coefficent of $x^n$

I’m having problems with proving Rodrigues’s formula. I’m stuck on expanding $$u=D^n((x^2-1)^n)$$ (where D is the differential operator) to “show that the coefficient of x^n in u is $$(2n)!/(n!)$$”. ...
2
votes
3answers
41 views

Find the coordinates of the points on the curve $y=2x^4 - 3x^2 + x - 7$ where the gradient is parallel to the line $y=3x$

I got that $\frac{dy}{dx}= 3$ (because of the gradient of the parallel line) and then, I found the derivative of the equation of the curve to be $8x^3 -6x +1= 3$. This simplifies as $8x^3 - 6x -2 ...