0
votes
1answer
30 views

Partial derivative of a Piecewise function

If I have the following equation: $$ f(x,y) = \begin{cases} x; & y \ge 0 \\ y; & y < 0 \\ \end{cases} $$ What are the partial derivatives (both x and y) of the function? I have trouble ...
10
votes
2answers
77 views

Second derivative of $f(f(\cdots f(x)\cdots )?$

For convenience, let's write $f_n(x)=f(f(\cdots f(x)\cdots )$ where $f$ is iterated $n$ times. Suppose: $$f(0)=0,\quad f'(0)=\alpha,\quad f''(0)=\beta$$ What is $f''_n(0)?$ I've found ...
0
votes
1answer
18 views

consequence of Mean Value Theorem

Let $f$ a continuous function on $[a, b]$ $a < b$ ,derivable on $(a, b)$ then there exist $c_1, c_2 \in (a, b)$ ,$c_1 \ne c_2$ such that $\frac{f (b) - f (a)}{b - a} = \frac{f '(c1) + f' ...
1
vote
1answer
27 views

Calculus - Trig Maximum Value Problem

When the rules of hockey were developed, Canada did not use the metric system. Thus, the distance between the goal posts was designated to be six feet. If Sidney Crosby is on the goal line, three feet ...
0
votes
0answers
13 views

Volume element notation

If $X=(x_1,x_2,\cdots,x_n)$. The notation $dX$ usually means $$dX:=(dx_1,dx_2,\cdots,dx_n)$$ or $$dX:=dx_1dx_2\cdots dx_n?$$ I am little bit confused about this, can anyone explain this for me? ...
0
votes
1answer
53 views

What is the exact meaning of Differentiability?

What is the exact meaning of Differentiability of a function at a given point? I know that If $\lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}$ exists, then we can say that the given function $f(x)$ is ...
0
votes
0answers
19 views

finding the equation of tangent to curve involving cos

I need help getting the equation of the tangent to the curve $y=3 \csc(2x)$ at $x=3\pi/4$. I used WA and got the derivative as $y'=-6\cot(2x)\csc(2x)$ I need to know how to get the derivative and ...
1
vote
2answers
21 views

implicit differentiating equation with $\cos$

I need help getting $\frac{d^2y}{dx^2}$ for $y−\cos y=2x$ Someone answered and got $(1+\sin y(x))3+4\cos y(x)$ but i was unable to follow their steps and didnt get how to do it. any HELP?
2
votes
3answers
64 views

Lagrange multipliers from hell

I was asked to solve this question, decided to try and solve it with lagrange multipliers as I see no other way: "Find the closest and furthest points on the circle made from the intersection of the ...
2
votes
1answer
30 views

Showing that if derivative is 0, function is constant ($f: U \rightarrow \mathbb{R}$ where $U \subset \mathbb{R}^n$)

Here's the question: Suppose that $f: U \rightarrow \mathbb{R}$ is differentiable on the open subset $U\subset \mathbb{R}^n$, and $Df(x) =0$ for all $x\in U$. Show that $f$ is constant on $U$. My ...
0
votes
0answers
8 views

prove where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable and is not differentiable [duplicate]

Classify where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable -- where it's not, prove it, and where it is, prove it. For $x\neq 0$ and $y\neq 0$, we can just treat it as $f(a,b) = \sqrt{a+b}$ (where ...
0
votes
2answers
54 views

Show where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable

Classify where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable -- where it's not, prove it, and where it is, prove it. My thoughts: For $x\neq 0$ and $y\neq 0$, we can just treat it as $f(a,b) = ...
0
votes
1answer
24 views

How to find the Frechet differential of a functional?

We know that the Fréchet differential $DF(u,\delta)$ of a functional $F:V\to V$ is satisfied (cf. Wiki) $$ \lim_{\delta\to ...
1
vote
0answers
25 views

Weierstrass function

I got stuck on this exercise from Prof. Tao's real analysis notes. Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be the function $$f:= \sum_{n=1}^\infty 4^{-n} \sin(8^n\pi x)$$ Show that for every 8-dyadic ...
1
vote
2answers
18 views

Derivative of sigmoid function

Sigmoid function is defined as $$\frac{1}{1+e^{-x}}$$ I tried to calculate the derivative and got $$\frac{e^{-x}}{(e^{-x}+1)^2}$$ Wolfram|Alpha however give me the same function but with exponents on ...
-4
votes
0answers
26 views

differentiate the given function. Simplify your answers [on hold]

In Exercise 1 through 28, differentiate the given function. Simplify your answers y=√2X
1
vote
1answer
24 views

solving the derivative of a function with cos

my question is y=cos^4(2x^2-1) here is my work `Dy/dx=4cos^3(2x^2-1) d/dx cos(2x^2-1) Dy/dx=4cos^3(2x^2-1) (-d/dx(2x^2-1)sin(2x^2-1)) ...
0
votes
1answer
22 views

Limits of Indeterminate Powers in Exponential Form using L'Hopital's Rule

I am trying to find the limit as $x \rightarrow 0$ of $x^x$ using L'Hopital's rule. I have written it in exponential form: $\lim\limits_{x \rightarrow 0} e^{x \ln x}$. I do not know how to put it in ...
0
votes
4answers
68 views

What is $\frac{1}{2} \int {e^{\frac{t}{2}}dt}$ equal to?

Would using substitution be helpful to get rid of the exponent of the variable? I tried substituting "$u$" in but it did not seem to help finding the integral.
1
vote
1answer
20 views

Single variable function derivative w.r.t. time?

I was studying calculus and I had doubts about this problem: (this is not homework) A circular wire expands due to heat so that its radius increases with a speed of $0.01 ms^{-1}$. How rapidly does ...
-2
votes
4answers
44 views

If $x^2 +xy =10$ then when $x=2$ what is $\frac{dy}{dx}$?

I solved for $y=3$ before I did the product rule and i'm not sure if that was the correct way to approach it.
0
votes
2answers
44 views

If $f(x)=x\sqrt{2x-3}$ what is $f'(x)$?

so far I re-wrote the problem using the product rule and chain rule so that i have $$\sqrt(2x-3)+x(\sqrt(2x-3)^{-1/2}$$ Now what?
0
votes
2answers
32 views

Find the parameter M

m(x+1)=e^|x| , m is a real number .Find the interval to which the parameter 'm' belongs , so that the previous equation has exactly two different solutions . Any idea how to approach this kind of ...
1
vote
4answers
59 views

How to differentiate $\frac{2x^5}{\tan x}$

$$\frac{2x^5}{\tan x}$$ I can differentiate $2x^5$ ($10x^4$) and $\tan x$ ($\sec^2 x$) but can't do that one Is there a rule I can apply?
1
vote
0answers
30 views

Total differentiation

For each of the functions below use the total diferential to approximate the change in $Y$ due to the given changes in $X$ and $Z$: $Y= X^2 + 4X -Z^2 -2XZ$, where $X=1$ and $Z = 4$ , and $\Delta X=2$ ...
0
votes
2answers
54 views

Why are the derivatives not treated the same?

It seems to me that derivatives are treated differently in certain places, but I do not understand why. Here is an example, if \begin{align} \frac{d}{dx} (\sqrt{1 + 4x^2}) & = \frac{1}{2\sqrt{1 ...
1
vote
3answers
71 views

How to differentiate this

$$e^{\tfrac{1}{\sin x}}$$ Help me how to differentiate that please help me Thanks.
1
vote
1answer
66 views

When are $\Delta x$, $\delta x$, $dx$, and $\text{đ}x$ exactly the same? When are they approximately the same?

As a follow-up to this related question, I'd like to know under what circumstances, if any, $\Delta x$, $\delta x$ and $dx$ all mean the same thing, and under what circumstances they can all be said ...
0
votes
1answer
21 views

Derivative rule question

In this image,from a website on compound interest derivations, why are you allowed to take the derivative of JUST the limit? Shouldn't you have to take the derivative of the lefthand side of the ...
0
votes
2answers
43 views

Higher Order Partial Derivatives

If i have 3 times differential function $ z= f(x^3 / y^4) $ how can i get: a) ${\partial z \over \partial x}$ b) ${ \partial ^2z \over \partial x^2}$ c) ${\partial^2z \over \partial x \partial ...
1
vote
1answer
19 views

Calculus rate of water filling a hemisphere

A large hemispherical wok has a diameter of 60cm. It is being filled at a constant rate of $50cm^3/s$. At what rate is the radius of the surface of the water increasing when the height of the water is ...
19
votes
2answers
278 views

When does $(uv)'=u'v'?$

In any calculus course, one of the first thing we learn is that $(uv)'=u'v+v'u$ rather than the what I've written in the title. This got me wondering: when is this dream product rule true? There are ...
1
vote
1answer
31 views

Question about finding where the function increases and decreases on $f(x)=\frac 1{x}$

$f(x)=\frac 1{x}, x\geq 1$ I have been staring at this equation for a bit. Things I'm confused on. the derivative of this is: $f'(x)= \frac {-1}{x^2}$ now, how am I supposed to find where this ...
0
votes
1answer
56 views

Find increasing/decreasing values of $f(x)= \frac{1}{2}(3x-1)$

$$f(x)= \frac{1}{2}(3x-1)\ \ \ \ x \le 3$$ I'm told I need to find where the derivative is increasing/decreasing. The problem is the $f'(x) =\frac{3}{2}$ so I'm not sure how to set this to zero to ...
2
votes
4answers
173 views

Derive an equation for derivative of ln x

$\frac{d}{dx}e^x = e^x$ use this fact together with the definition of the natural log $\ln x$ as the inverse of the function of $e^x$ to derive an equation for the derivative of $\ln x$.
1
vote
0answers
17 views

Is the variance of the left truncated normal distribution decreasing in lower bound?

I am wondering whether the variance of the left truncated normal distribution is always decreasing in $\alpha$ (lower bound)? The untruncated distribution of x is $\mathcal{N}(\mu,\sigma^2)$. The ...
1
vote
3answers
26 views

Find the slope of a tangent to a curve when $x = 4$

I am being asked to find the slope of a tangent to a curve when $x=4$. The equation I have is $f(x) = 4x^3 - 5x + 2\sqrt{x}$ I'm a beginner and I must say that I'm having a hard time with this. I ...
0
votes
2answers
22 views

A basic question related to differentiability?

Was studying the continuity and differentiability,but after so many attempts to understand derivative still dont get what it really is.Let me understand this question? I have a function say $f(x)=x^2$ ...
1
vote
0answers
37 views

Show that this is not differentiable at any point in $\mathbb{R}$

Define $\phi: \ \mathbb{R} \rightarrow \mathbb{R}$ by $$ \phi(x) = \begin{cases} x\ :\ 0\le x\le \frac{1}{2}\\ 1-x :\ \frac{1}{2} \le x \le 1 \end{cases}$$ And then extend ...
1
vote
2answers
59 views

Let $f(x) := x^2 \sin \frac 1 x, f(0)=0$. Show $f$ is differentiable on $\mathbb R$.

Let $f: \mathbb R \rightarrow \mathbb R$ defined by $$f(x) := \begin{cases}x^2 \sin \frac 1 x\ & x \neq 0\\ 0\ & x = 0\end{cases}$$ Show $f$ is differentiable on $\mathbb R$: Let $\epsilon ...
1
vote
1answer
39 views

Need help with implicit differentiation

hi i need help on finding the $\dfrac{d^2 y}{d x^2}$ for $x^6-y^6=14$ i got $$\frac{5x^4(y^6-x^6)}{y^{11}}$$ but im not sure if its right or not also i am completly stuck on getting $\dfrac{d^2 ...
4
votes
0answers
74 views

A tough one: show that this is not differentiable at any point in R

Here's the question: Define $\phi: \ \mathbb{R} \rightarrow \mathbb{R}$ by $$ \phi(x) = \begin{cases}x & 0\leq x\leq\frac{1}{2}\\ 1-x & \frac{1}{2}\leq x\leq 1\end{cases}. $$ And then ...
2
votes
1answer
24 views

Is there a set formula for integration like there is for derivatives?

I know that the derivative of $f(x)$ must be $$f'(x)=\lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ We can use this formula to derive the derivatives of some functions like $\sin(x)$. Is there such a ...
3
votes
1answer
45 views

Problem with differentiable function: is it concave up when the derivative is increasing?

This makes sense to me, and I feel like it would be an easy argument IF I could use the second derivative. I'm only given that f is differentiable, NOT twice differentiable. Any help?
0
votes
1answer
39 views

show that $f(x,y) =2x^2 + 3y$ is differentiable at $(0,0)$ by finding a linear function T

Here's the question: Prove that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined $f(x,y) = 2x^2 + 3y$ is differentiable at $\begin{bmatrix} 0\\0 \end{bmatrix}$ by producing a linear function T and ...
0
votes
0answers
36 views

Calculus and Matrices

Suppose I have a linear operator $T: \mathbb{R} \rightarrow \mathbb{R}$, and also suppose that it's a composition of elementary functions, so its derivative, $T'$, is reasonable easy to find. I can ...
1
vote
2answers
40 views

How to differentiate $f(x) = 1-xe^{1-x}$ w.r.t. $x$?

I would like to differentiate the following with respect to $x$: $$f(x) = 1-xe^{1-x} \tag 1$$ How would I do this please? I can see that the 1 would disappear, then I am left with ...
2
votes
1answer
56 views

Find the $dy/dx$ of $y=y=x\int \limits_2^{x^2}\sin\left(t^3\right){d}t$

Need to find $\frac{dy}{dx}$ for this: $$y=x\int \limits_2^{x^2}\sin\left(t^3\right){d}t$$ I tried using the chain rule and I am still left with $\int \limits_2^{x^2}\sin\left(t^3\right){d}t$ in my ...
2
votes
1answer
31 views

Integral of $e^{(a+ib)x}$

Given the function $f:\mathbb{R}\rightarrow \mathbb{C}$, such that $f(x)=e^{(a+ib)x}$, how can I compute $f'(x)$ and $\int f(x)dx$ ? Certanly, one can use the identity $e^{ibx}=\cos(bx)+i\sin(bx)$ and ...
0
votes
1answer
14 views

Derivatives with ln Issues

I got 3x^2/x^3-7 but I'm not sure where to go from there. Also I ran into this problem and haven't been able to figure it out. Thanks for your time, I really appreciate it.