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

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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 $\Delta X=2$ ...
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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 ...
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59 views

How to find derivative of an integral

So I am given $$\frac{d}{dx}\left(\int_0^x e^{t^2}\ dt\right)$$ How would I go about solving this?
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Application of related rates

Here is a question from a sheet my math teacher assigned me. ...
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20 views

Maximize the directional derivative

Find the points $(x,y)$ and the directions for which the directional derivative of $f(x,y)=3x^2+y^2$ has its largest value, if $(x,y)$ is restricted to be on the circle $x^2+y^2=1$. For the point ...
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71 views

How to differentiate this

$$e^{\tfrac{1}{\sin x}}$$ Help me how to differentiate that please help me Thanks.
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1answer
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Directional Derivative and differentiability

My question is similar, but not equal to this...Question on linearity of directional derivative Let $f'_{h}(a)$ be the directional derivative. And for the function $f:\mathbb{R}^n\rightarrow ...
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106 views

Prove $f $ is identically zero

$f:\Bbb R \to\Bbb R $ is differentiable, $f(0)=0$ and $|f'(x)|\le|f(x)|$ for all $x$ then prove $f$ is identically zero. I tried to use mean value theorem and end up in $|f(x)|\le |x||f(c)|$ for some ...
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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 ...
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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 ...
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1answer
28 views

$\frac{d(X'X)}{dX}=?$

Thanks a lot for reading my thread. I am wondering what is the derivative of $X'X$ with respect to $X$? Here $X$ is a vector/matrix, and $X'$ is the Hermitian matrix of $X$; It would be great if ...
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41 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 ...
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14 views

Second order of total differential of function:

If i have function $ z(x, y)= x^2 + e^{x*y} -y^3 $ how can i find 2nd order total differential? Can someone explain me step by step please.
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1answer
14 views

Least Squares Curve fitting, why make the error derivative 0?

My notes define the error of least squares approximation as: $$ E=\sum_{i=1}^n(y_i-f(x_i))^2\tag1 $$ Which, for a straight line gives: $$ f(x_i)=a+bx\tag2$$ $$ E=\sum_{i=1}^n(y_i-(a+bx_i))^2\tag3 ...
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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 ...
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1answer
21 views

Use Differentiation to fine the absolute minimum and absolute maximums

Find the absolut maximum and absolute minimum values of the function f(x)= 4x/8x+4 On the interval [3,7] I'm quite lost on this question, if someone can work through it completely so i have a ...
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1answer
21 views

Implicit Differentiation to find equation if the tangent line to a curve

Use implicit differentiation to find an equation of the tangent line to the curve: $x^2+y^2=(2x^2+2y^2-x)^2$ At the point: $(0, \frac{1}{2})$ Hi I'm really lost with this question, can somebody ...
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Differentiation to find the tangent of a lines equation.

Find an equation of the tangent line to the curve $y = e^x \sin x$ at the point $(0,0)$. Tangent Line Equation=? I know i must differentiate the equation but then what? Can somebody please work ...
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26 views

Differentiation Problem solving

A certain magical substance that is used to make solid magical spheres costs $\$800$ per cubic foot. The power of a magical sphere depends on its surface area, and a magical sphere can be sold for ...
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255 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 ...
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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 ...
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1answer
22 views

Definition of Derivative for $sgn(x)$

When using the definition of the second derivative for $sgn(x)$, I'm a little confused on evaluating something like $sgn(x+h)$. Since $h\rightarrow 0$ does that mean that I should treating $sgn(x+h)$ ...
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55 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 ...
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Multiplication of two derivatives

I am trying to figure out the following step in a long derivation: $$ \frac{dV}{dS} \, \frac{dV}{dx} \, \frac{1}{S} = \frac{d^2V}{dx^2} \, \frac{1}{S^2} - \frac{dV}{dx} \frac{1}{S^2}. $$ Does this ...
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(newbie) spectral derivative

I have data that form a scalar field on a 2D grid, evenly spaced. The grid has a finite size. There is no particular periodicity patern in my data. I want to calculate the value of the gradient at ...
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Linear operator exists then differentiable?

Let $E_{\text{open}} \subseteq \mathbb{R}^n$, and let $\vec{x_o} \in E$. Let $\vec{f}: E \rightarrow \mathbb{R}^m$. If there exists a linear operator $A: \mathbb{R}^n \rightarrow \mathbb{R}^m$. such ...
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1answer
26 views

Linear transformation from $R^2$ to $R^2$.

Let $\vec{f}: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, where $\vec{f} (\vec{x}) = (x+y^2, x^3+5y)$ and $\vec{x} = (x,y) \in \mathbb{R}^2$. Let $\vec{h} = (h_1, h_2)$ and $\vec{a} = (1,1) \in ...
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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 $\ln x$.
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23 views

If $f$ is continuous on $(0,5)$, is it uniformly continuous on same interval

I have a function, $f$ that is differentiable on $(0,5)$, and I know it is continuous on $(0,5).$ Is it also uniformly continuous on $(0,5)?$ I believe it is. I now know that it is not. Can someone ...
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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 ...
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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 ...
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1answer
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Uniform convergence and differentiation proof of remark 7.17 in Rudin's mathematical analysis

Rudin page 152 Theorem 7.17: Suppose $\{f_n\}$ a sequence of functions, differentiable on $[a,b]$ and such that $\{f_n(x_0)\}$ converges for some point x0 on [a,b]. If {f′n} converges uniformly on ...
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Why are differential of $\sin^2(x)$ and integral of $\sin(2x)$ not the same?

I was working on a list of common integrals and differentials and I came across this question. If $${d\over d\theta}(\sin^2\theta) = \sin(2\theta)$$ Then why is $$\int \sin(2\theta) \space d\theta = ...
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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 $f(x)=x^2$ ...
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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 ...
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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 ...
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Lie Group - derivatives

This is really a simple question. Let $A$ be an associative, nilpotent real algebra, and set $[a,b]=ab-ba$, define the exponential map as usual, that is $exp(a)=1+a+\frac{a^2}{2}+...$. Let ...
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28 views

The minimum distance from the circle $x^2+(y+6)^2=1$ to parabola $y^2=8x$?

What are the coordinates of the points on the parabola $y^2=8x$ which are at the minimum distance from the circle $x^2 + (y+6)^2=1$?
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Derivative of an integral where the variable of derivation is in the integral

How does one deal with finding $$\frac{d}{dx} \int_{0}^{x} x \, dt$$ or any integrand function that is a constant relative the integration but a variable in the derivation? Thanks.
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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39 views

Why does $d$ mean?

What do the $d$'s mean? I've seen them in other formulas as well.
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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 ...
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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 ...
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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 ...
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questions about Darboux theorem

i saw a proof of Darboux theorem. it says that we have function f and her derivative is f ', and f matches all the condition of Darboux theorem. but than, it says that if f '(a)=f '(b) then the claim ...
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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 ...