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

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1answer
24 views

Differentiation proof

Find the co-ordinates of the point on a curve $y=x^2+3x-1$ at which it is parallel to the line $ y=5x-1?$ unsure how to solve this
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1answer
29 views

Vector Cross product - Rearranging issue

Given Data in question I have following relations in vector space$\begin{eqnarray}n_0^{'}(s)=-\kappa(s) \times n_0(s)\\n_1^{'}(s)=-\kappa(s) \times n_1(s)\\n_2^{'}(s)=-\kappa(s) \times ...
0
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1answer
53 views

For what values of $r$ does the function $y = e^{rx}$ satisfy the differential equation $y'' − 4y' + y = 0$?

I took $y'' − 4y' + y = 0$ and substituted $e^{rx}$ for $y$ but then I just get $e^{rx}=0$ and that's where I'm stuck, if that's even along the correct line of thought. For what values of $r$ does ...
0
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1answer
28 views

Differentiating a function and using the result to calculate the indefinite integral of another.

We should differentiate the function $f(x) = \sqrt{cosx}$ and use the result to calculate the indefinite integral $\int \frac{sinx}{\sqrt{cosx}}dx$. So I started by differentiating $f(x) = ...
2
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1answer
21 views

Expression of the Runge function's derivative

I am trying to get the nth derivative of the Runge function i.e. i want : $$\dfrac{d^n}{dx^n} \dfrac{1}{1+25x^2}.$$ Mathematica gives me the answer : $$\dfrac{d^n}{dx^n} \dfrac{1}{1+25x^2}=\dfrac{n! ...
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0answers
31 views

Integration by Parts in matrices

Given Data in the question We have a given equation based on matrices as follows $\frac{\mathrm{d} R(s)_{3\times3}}{\mathrm{d} s}=R(s)_{3\times3}K(s)_{3\times3} \tag 1$ $\frac{\mathrm{d} ...
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4answers
27 views

Find the intervals on which $f(x) = 8\cos 4(x)$ decreases for $0 \le x \le π $?

Find the intervals on which $f(x) = 8\cos 4(x)$ decreases for $0 \le x \le π $. What is the fast way to compute it?
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4answers
45 views

Evaluating $g'''(\pi/4)$, given $g(x) = \sec (x)$

Given the function $$g(x)=\sec(x)$$, I have to solve for $g'''(\pi /4)$. I calculated the 3rd derivative to be $$ g'''(x)=\sec x\tan ^3 x+5\sec ^3x \tan x$$ I just don't know how to evaluate the ...
7
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1answer
62 views

Evaluating $\int x^n e^{x}dx$

I consider, for $n=0,1,2,...$, $$ u_n(x)=\int x^n e^{x}dx.$$ I've performed an integration by parts giving $$ u_n(x)=nx^{n-1} e^{x}-nu_{n-1}(x).$$ I'm looking for a closed form. Thank you.
3
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2answers
50 views

Find the value of $f_{xy}$ at the point (0,0)

Let f be the function defined for all (x,y) as follows: $f(x,y)= \begin{cases} \frac{xy(x^2-y^2)}{x^2+y^2}, &\text{if }(x,y)\ne(0,0)\\ 0, &\text{if }(x,y)=(0,0) \end{cases}$ What is the ...
0
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2answers
41 views

How to take this derivative

My question is straightforward: I need to evaluate an expression of the form $$ \frac{\partial}{\partial t}\sum_{k=0}^{t}\varphi(k,t) $$ How is this done, usually?
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1answer
28 views

Determining whether a function of two variables is continuously differentiable

I'm just trying to solidify the different cases for functions When dealing with functions that are $C^1$, say I have some function: $$f(a,b)=\begin{cases}(a^4+b^4)/(a^2+b^2)\quad &\text{ if } ...
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2answers
44 views

Height and velocity of ball thrown vertically

A ball is thrown upward from roof of 32 foot building with velocity of $112$ ft/sec. The height after $t$ seconds is: $s(t)=32+112t-16t^2$. (a) Find the maximum height that the ball reaches. ...
2
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1answer
72 views

A property of homogeneous of degree p functions:

Prove that if $f(x_1,...,x_n)$ is homogeneous of degree $p$, i.e; $f(tx)=t^pf(x)$. Then: $$(x_1 \frac { \partial}{\partial x_1} +...+x_n \frac { \partial}{\partial ...
1
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1answer
25 views

Writing The Derivative Of $f(x)$ With Respect To $g(x)$ In Limit Form

What would be the proper way to represent this derivative in the limit form? $$\frac{\mathrm{d} }{\mathrm{d} g(x)}[f(g(x))]$$ In my attempt to solve this I've tried to word out the derivative: The ...
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0answers
28 views

Complex Derivative of z^m

I'm asked to show that d/dz(z^m)=mz^(m-1) using the Cauchy Integral formula (for derivatives) and a binomial expansion. I've tried writing out d/dz(z^m) using the CIF (for derivatives) and writing ...
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2answers
30 views

Differentiation with surds

I know dy / dx = n^n-1 I have the problem $y = \sqrt{x^2+2x}$ I have broken that down to $y = \sqrt{x^2 +2x}$. The x would differentiate to 1, but how do you differentiate the surd?
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2answers
37 views

Taking the nth derivative as a function as n becomes very large (Taylor series)

Suppose I have a function: $F(x) = \frac{1-x^2}{1-x^2-2x^3}$ How would I go about approximating the $n-th$ derivative of $F(x)$ when $n$ becomes very large? Edit: Motivation: I ask because I want ...
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0answers
40 views

Simple exercise in analysis

Suppose that $a,b\in \mathbb{R}, a<b$ and $X$ is some Banach space. We have a map $$ F: \mathbb{R}^2 \supset(a,b)^2\ni (x,y)\mapsto F(x,y)\in X. $$ Suppose additionally, that for each fixed ...
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2answers
25 views

Derivative problem to minimum building cost

Let point S to P is x km. so cost between P to O should be [(10^2)+x^2]*2 S to P is x*1 million and P to R is (10-x)*1 So total should be [(10^2)+x^2]*2 + x*1 + (10-x)*1 ? Should I find the ...
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0answers
26 views

Derivative problem with inflection point and end points

Not quite sure about that, so overall this is a positive slope graph right? And at middle point of time (concavity) will have significant increase of depth, and the inflection point is here right? ...
3
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3answers
50 views

Deriving $\frac{8}{\sqrt{x-2}}$

I'm not sure how to derive this: $$\frac{8}{\sqrt{x-2}}$$ I tried $$8 \cdot \frac{1}{\sqrt{x-2}}$$ $$8 \cdot (\sqrt{x-2})^{-1}$$ Differentiating w.r.t. $x$, $$8 \cdot -1 \cdot ...
3
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5answers
95 views

The derivative of something with respect to $3x+5$?

If you take $(3x+5)^2$ and differentiate it with respect to $3x+5$ it's just $2(3x+5)$. Can someone explain to me how this would actually work out? I understand normal derivatives with respect to say, ...
2
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1answer
140 views

What Notation Do I Use To Fix Ambiguity Writing Chain Rule

I'm a calculus noob learning over the internet. I think the best way to ask my question is just to put up a little diagram I made in paint. Now this is my attempt to write the chain rule using d/dx ...
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2answers
30 views

Find equation of tangent line, derivatives

if $g(x)= xf(x)$, where $f(3)=4$ and $f'(3)=-2$, find an equation of the tangent line to the graph of $g$ at the point where $x=3$ The help would be appreciated Thanks!
0
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3answers
44 views

Chain rule clarification?

I am not understanding how if you have some function that is f(g(x)) that the derivative of that function is f '(g(x)) g '(x). This doesn't make sense to me because shouldn't f'(g(x)) be f'(g(x))? I ...
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2answers
32 views

Building a cubic polynomial with certain parameters

I am trying to build a cubic polynomial $y = ax^3 + bx^2 + cx + d$ with the following conditions: $y(0) = 3$ $y'(0) = 2$ $y'(1) = 0$ $y(1) = 4$ Unfortunately, solving the required system yields ...
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0answers
26 views

Schwarzian Derivative

Determine the Schwarzian derivative of the following function: a.$$ f(x)= ax^2 + bx +c$$ b.$$ g(x) = x^3 + (1/2)x$$ c. $$h(x) = x^n $$ with n greater than or equal to 3 so I got the following ...
3
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3answers
40 views

Derivative of a function with respect to another function.

I want to calculate the derivative of a function with respect to, not a variable, but respect to another function. For example: $$g(x)=2f(x)+x+\log[f(x)]$$ I want to compute $$\frac{\mathrm ...
0
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1answer
22 views

Logistic regression maximum likelihood derivation

the following equations are given: $\sum_{j=1}^c\hat{P}_j = 1$ $\sigma_i(\mathbf{z}; \theta) = \frac{exp(\mathbf{\theta}_i^T\mathbf{z})}{\sum_{j=1}^cexp(\mathbf{\theta}_j^T\mathbf{z})}$ $L = ...
0
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1answer
21 views

question about derivative with respect to the log of the parameter

I have a function $K(\phi)$ and I need to compute $\frac{dK}{d \log(\phi)}$. I am guessing I can break this as: $$ \frac{dK}{d \log(\phi)} = \frac{dK}{d\phi} \frac{d \phi}{d \log(\phi)} = ...
0
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2answers
20 views

Derivative of simple fraction

What is the fastest way of finding the derivative of: $\frac{x}{x+K}$ (simplified form) is there a substitution I miss or does the quotient rule the job here? There should be a quick way of finding ...
1
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1answer
41 views

The derivative as a linear function

In Milnor's Topology From the Differentiable Viewpoint, the derivative of a smooth map $f: U\to V$ is defined as $$ \mathrm{d}f_x: \mathbb{R}^k \to \mathbb{R}^l $$ $$ h\mapsto \lim_{t\to0} ...
0
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1answer
26 views

Compute $\frac{dy}{dx}$ as a function of $y$ when $x=\cos y$

The function $\cos^{-1}(x)$ is defined for $-1\leq x\leq1$ and $0\leq y\leq \pi$ by the equivalence $$y=\cos^{-1}(x)\Leftrightarrow x=\cos y$$ (a) Compute $\dfrac{dy}{dx}$ as a function of $y$ by ...
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2answers
13 views

derivative of function of T

How do I take the derivative of: $\left ( \frac{1}{T^4} \right ) \left (\frac{1}{K-T} \right )$ Can I just use the product rule? IT seems like it get pretty complicated pretty fast
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0answers
25 views

“Angle-preserving” equivalent to conformal?

I'd like to investigate the common turn of phrase that conflates "angle-preserving map" with "conformal map". Let $f:\Bbb R^2\to\Bbb R^2$ be a continuous function. I'll define $f$ to be ...
1
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1answer
34 views

Why $(-1 \cdot h) = -1$ when $h$ approaches $0$?

I'm starting to learn about derivatives. I have an example, but I'm not sure about one point in it. I'm pretty sure it relates to basic limit knowledge: Derive $$\frac{x}{x-1}$$ So the ...
0
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1answer
23 views

Derivative of f(x, y, z(x, y))

I have computed the derivatives to be: $dw/ds = (-5y+5z)(t) + (-5x-2z)(e^{st}(t))$ $dw/dt = (-5y+5z)(s) + (-5x-2z)(e^{st}(s)) + (-2y + 5x)(2t)$ I calculated the first answer by evaluating dw/ds ...
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3answers
41 views

Help with understanding proof of the product rule

$F(x)=f(x)g(x)$ $\lim_{h \to 0}\dfrac{f(x+h)g(x+h)-f(x)g(x)}{h}$ Then, the notes I'm reading say: "the numerator $f(x+h)g(x+h)-f(x)g(x)$ is a difference that involves $x$ changing to $x+h$ for ...
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0answers
22 views

Get the critical points and find the máximum or mínimum of $f(x,y,z) = (x^{2} + 2y^{2} +1)\cos{z}$

I'm trying to solve this problem: Get the critical points and find the máximum or mínimum of $f(x,y,z) = (x^{2} + 2y^{2} +1)\cos{z}$ First, I founded the gradient: $\nabla f(x,y,z)= (2x\cos{z}, ...
0
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1answer
21 views

Change of Variable involve derivative

Let me just give the 1-D version of my problem. Let $u\in C_c^\infty(R)$ and define $u_r(x):=u(rx)$. Then I am trying to evaluate the integration $\int_R u_r'(x)dx$. Here is my steps: $$\int_R ...
0
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1answer
39 views

Very basic question about the definition of the derivative

Why is the definition of the derivative shown here as $\dfrac{\Delta x}{\Delta y}$ if immediately above the slope (derivative) is defined as $\dfrac{\Delta y}{\Delta x}$? Why is $\Delta x$ in the ...
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0answers
10 views

How to apply the chain rule inside the expectation operator?

I'm working with a function that involves a term $$ E_t\left[ V(w_{t+1})^{1-\gamma}\right]^{\frac{1-\eta}{1-\gamma}} $$ I need to differentiate this with respect to $w_{t+1}$. How should I apply the ...
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0answers
40 views

On what value of $f(X)$ minimizes $E[(Y-f(X))^2|X]$

$X$ and $Y$ are random variables. The question is: what value of $f(X)$ minimizes $E[(Y-f(X))^2|X]$. I am pretty sure I have found the solution to this problem by writing: $$E[(Y-f(X)-E[X|Y] +E[X|Y] ...
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0answers
12 views

Change of variables in Lagrangian

Question Let $\psi : [t_0, t_1] \to \mathbb{R}$ be a smooth function such that for $t \in[t_0, t_1], \dot{\psi(t)} > 0$ and also so that $\psi(t_0) = x_0$ and $\psi(t_1) = x_1$. Using the ...
0
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1answer
49 views

Integral using Beta Function and Gamma Function

Interestingly, I seem to have an integral I have posted before, but I want to take a different approach to it. $\int_{0}^{1} \frac{\ln(1+x)}{1+x^2} \,dx$ The beta function states, $B(x,y) = ...
3
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2answers
41 views

How to find the derivative of $(3x-1)^2(2x+3)^2$

I used the power rule and the chain rule and ended up with this: $$y'= (3x-1)^2 \times 2(2x+3) \times 2 + (2x+3)^2 \times 2(3x-1)\times 3$$ The next step, which I do not understand how it is combined ...
0
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4answers
41 views

derivative of $f(x)=(4x^2+9)^7(7x^2+3)^{12}$

$f(x)=(4x^2+9)^7(7x^2+3)^{12}$ I used the product rule and came up with: $y'=4(x^2+9)^7(168(7x^2+3)^{11})+56x(4x^2+9)^6(7x^2+3)^{12}$ Why is this wrong?
1
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4answers
60 views

Derivative of $f(x)=-6\sin^4 x$

$f(x)=-6\sin^4x$ $f(x)=-6\sin x^4$ $f'(x)=-6\cos x^4(4x^3)=-24x^3\cos x^4$ What am I doing wrong? Please show the steps.
1
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1answer
28 views

Derivative Functions [closed]

Consider $f(x)= ax^2 + bx$ where $a$ and $b$ are real numbers. If $f(1)=-1$ and $f'(-1)=-7$, find the values of $a$ and $b$? I genuinely do not understand how to do it! Please help