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

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Derivative tests question

Show that $k(x) = \sin^{-1}(x)$ has $0$ inflections $2$ critical points $0$ max/min I find that the first derivative is $$\frac{1}{\sqrt{1-x^2}}$$ Second derivative is ...
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1answer
68 views

Definition of a 2-variable function derivative

I read this definition in a book of multivariable calculus: $f(x,y)$ is differentiable at $(x_0,y_0)$ if it can be expressed as the form $$f(x_0+\Delta x, y_0+\Delta y)=f(x_0,y_0)+A\Delta ...
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1answer
39 views

Differentiability of the function $x \mapsto |x|^{3/2}$ at $x = 0$

Could someone please explain whether the function $$\vert x \vert^{3/2}$$ is differentiable at zero? ($x$ here is a real number.) I tried investigating the right and left-sided limits (i.e., the ...
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1answer
22 views

The existence of the $n$th derivative at $c$ presumes the existence of the $(n-1)$st derivative in an interval containing $c$

The following is from Introduction to Real Analysis by Bartle. If the derivative $f'(x)$ of a function $f$ exists at every point $x$ in an interval $I$ containing a point $c$, then we can consider ...
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3answers
62 views

Calculating arc length $y=x^2$

I picked this example for practice and got stuck with it. Someone moderate me if I am in the right path. I need to calculate the length of arc s, on the section of the curve $y=x^2$ with $0≤x≤1$ My ...
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1answer
24 views

Differentiating hyperbolic functions.

$\DeclareMathOperator{\sech}{sech}$Can anyhow me how to differentiate the following? I already tried using the product rule, but I can't quiet seem to succeed. $\sech^{2} x$. $2\bigl(\cosh(2x) - ...
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1answer
32 views

Differentiability implies continuity in $R^2$

Let F be a function from $R^2$ to $R^2$. F is differentiable at a point (a,b) in $R^2$, prove that F is continuous at this point. Can i write F(x,y)= F(a,b)+ c(x-a)+ d(x-b)+e where c,d,e are real ...
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66 views

Finding slope at a point in a direction on a 3d surface

This is not a duplicate, I have attempted the question and am not sure why my answer is incorrect. QUESTION: The surface with equation $z = x^3 +xy^2 $ intersects the plane with equation $ 2x−2y = 1$ ...
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1answer
42 views

Showing differentiability for a multivariable piecewise function

Let $$f(x,y)=\begin{cases} xy\sin(x/y) & y\neq 0 \\ 0 & y=0\end{cases},$$ show whether $f(x,y)$ is differentiable at $(0,0)$. It seems that there are multiple ways to do this but ...
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33 views

Find the absolute min and max in the given intervals

$k(x) = e^{-\frac{x^2}{2}}$ on $[-1,2]$ I think the derivative of that is $ -x e^{-\frac{x^2}{2}}$. I don't know how to find zero from that equation.
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56 views

Differentiation of the function $\operatorname{li}(x) = \int_2^x \frac{dt}{\ln(t)} $

I have to differentiate with respect to x: $$\operatorname{li}(x) = \int_2^x \frac{dt}{\ln(t)} $$ I havn't come across this before, so my idea is to integrate it first? (Backward right?). let ...
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2answers
32 views

Computing the limit of an integral (Derivatives of Integrals)

Assuming that $f(x)$ is continuous in the neighborhood of $a$, compute $$ \lim_{x \to a} \frac{x}{x-a} \int_a^x f(t)dt $$
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31 views

How can I approximate a function that is not derivable with derivable ones?

Suppose that I have a function whose graph has many angles (i.e. my function is not derivable). How can I approximate this function with derivable ones? Thank you!
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40 views

Is there a locally-distance-preserving map projection?

I'm trying to figure out if there is a family of map projections which preserve local distances: in other words a family of functions $f \in S^2 \rightarrow K, K \subseteq \mathbb R^2$ such that for ...
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1answer
29 views

Finding $y$ In Calculus(Area) Problem? [duplicate]

Find the number b such that the line $y=b$ divides the region bounded by the curves $y = x^2$ and $y = 4$ into two regions with equal area.
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113 views

How to solve this differential equation please?

I'm trying to solve: $$\frac{dz}{dx}+2xz=2x$$ I have got the integrating factor as $$e^{\int 2x dx}=e^{x^2}$$ and so $$ze^{x^2}=\int {2xe^{x^2}} dx+ C$$ But I don't know how to proceed it's mainly ...
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89 views

Finding the rate of rising water.

Water is pouring into a conical tank at a rate of 8 cubic feet per minute. If the height of the tank is 12ft, and the radius of its circular opening is 6ft, how fast is the water level rising when ...
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3answers
47 views

Show differentiability at a point then find differential where $f(x,y) = (x^2, xy+y^2)$

Show differentiability at a point then find differential where $f(x,y) = (x^2, xy+y^2)$ Want to show that $f(x,y) = (x^2, xy+y^2)$ is differentiable at $(a,b)$ and then calculate the differential ...
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1answer
14 views

By using the Chain Rule, find d$g(u_0,v_0)$ where $f(x,y) = x^2 + xy - y^2$ and $g(u,v) = f(u^2,uv)$

By using the Chain Rule, find d$g(u_0,v_0)$ where $f(x,y) = x^2 + xy - y^2$ and $g(u,v) = f(u^2,uv)$. $(x,y) \in \mathbb{R^2}$ and $(u,v) \in \mathbb{R^2}$ My Thoughts So I understand that $g(u,v) ...
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2answers
49 views

Questions about derivatives with e and ln in them

When taking the derivative of e^x I was under the impression it remained as e^x, but some books I've been reading have confused me.They show things like e^ -x and take the derivative as I mentioned ...
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30 views

is F differentiable at x0,y0

$F(x,y)=(x^2-y,xy)$, how to show that F is differentiable at $(x_0,y_0)$ and find $dF(x_0,y_0)$ My attempt: I think I understand part 2 of this question, $dF=\begin{pmatrix} 2x & -1 \\ y & x ...
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1answer
79 views

first order approximation of scalar function of matrix ( Mahalanobis distance)

I have tried to compute the 1st order approximation using Taylor's expansion of the Mahalanobis distance: $f(\mathbf{X})=\mathbf{a^TXa}$, where $\mathbf{a}\in \mathbb{R}^N$. The function maps ...
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18 views

Polylogarithm and unclear statement

I am trying to solve this question which may not have an answer at all, but any clarification would be much appreciated. I also tried to explain what I have tried/thought about it below. Let ...
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1answer
27 views

“Symmetric” numerical computation of second derivative

When numerically computing a first derivative, it is better to use $$f'(x) \approx \frac{f(x + \Delta x / 2) - f(x - \Delta x / 2)}{\Delta x}$$ than to use $$f'(x) \approx \frac{f(x + \Delta x) - ...
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1answer
24 views

Implicit Differentiation Solution Verification [Inverse Trig Function]

Find $\frac{dy}{dx}$. $$ \\ \\ \text{ } \\ \arctan{y^3} = \sin^3{x} + \cos^3{(yx)} \\ \text{ } \\ y^3 = \tan{(\sin^3{x} + \cos^3{(yx)})} \\ \text{ } \\ \frac{d}{dx}[y^3] = ...
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1answer
43 views

Proving inequality

Let $f$ be a twice differentiable function and let M, N, and P be the least upper bounds of |$f$(x)| |$f'$(x)| and |$f''$(x)| respectively prove that the square of N can never exceed 4 MP. I thought ...
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4answers
65 views

Differentiate $\sqrt{1+f(x)^2}/(1+f(x))$

Is there a trick I'm overseeing? I have $\frac{\sqrt{1+f(x)^2}}{1+f(x)} = \sqrt{1+f(x)^2} \cdot (1+y)^{-1}$ . First differentiation: $$[\sqrt{1+f(x)^2} \cdot (1+y)^{-1}]' = ...
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54 views

Sum of derivatives

Let $$f(x)=\frac{x}{e^x−1}$$ if $x≠0$, and let $f(0)=0$. Let $f^{(n)}$ denote the $n^{th}$ derivative of $f$. Then find the sum $$f^{(1)}(0)+f^{(3)}(0)+f^{(5)}(0)+\ldots$$
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70 views

The limit needed to find the derivative of e^sin(x) at x=1.

I'm trying to figure out why this: is the answer to this problem: What are the steps that take the numerator from $e^{\sin(1+h)} - e^{\sin(1)}$ to $e^{\sin(h)} - 1$?
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24 views

Find $a,b,c$ of polynomial function (Hermite interpolation)

Given abscissae $x_1 < x_2 < \dots < x_N$ and corresponding data values $\{y_i\}_{i=1}^N$ and derivative values $\{y_i'\}_{i=1}^N$, consider the following Hermite interpolation method: For ...
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2answers
57 views

Find values of b for which $f(x)=x^3+x^2+bx+6$ is increasing for all values of $x$

For the function defined by $f(x)$, find the values of $b$ that results in $f(x)$ increasing for all values of $x$. I found the derivative: $f'(x) = 3x^2+2x+b$ and I know that it should always be ...
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1answer
34 views

Use the definition of differentiation on a piecewise function.

I need to find the derivative at $x=0$. $$ f(x)= \begin{cases} x^2\sin(1/x) & \text{if } x\neq 0 \\ 0 & \text{if } x \leqslant 0 \end{cases} $$ Using the definition, I know that it's equal ...
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41 views

Explain why the chain rule is needed to find derivatives if $y$ is a function of $x$?

I know that if $y$ is a function of $x$, or $y=f(x)$, you need to use the chain rule to find it's derivative. Let's say I want to find the derivative of $y^2$ and $y$ is a function of $x$. ...
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2answers
116 views

Find the exact length of the arc of this curve

$y = 2e^x + (1/8)e^{-x}$ ... on the interval $[0, \ln(2)]$ I know am supposed to user the Arc Length formula, but I'm not sure if I have the derivative of this function correct. I came up with: ...
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78 views

Show that the cubic equation has one real roots

Show that $x^3+ax+b=0$ has a) only one real root when $a>0$ b) at most only one of it's roots are in $(-\sqrt{-a/3},\sqrt{-a/3})$ when $a<0$. For a) I supposed that it had two real roots ...
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36 views

Implicit logarithmic differentiation to find the horizontal tangents of an exponential function

The graph of $y = 6{(3{x}^2)}^x$ has two horizontal tangent lines. Find equations for both of them. $$ \\ \begin{align} \\ y &= 6{(3{x}^2)}^x \\ y &= 6 \cdot {3}^x \cdot {x}^{2x} \\ \ln{y} ...
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48 views

Show that the inverse function to $f(x)=\int_{1}^{x}\frac{dt}{t}$ is differentiable

Show that the inverse function to $$f(x)=\int_{1}^{x}\frac{dt}{t}$$ is differentiable. I know that the integral is $\ln(x)$, but I don't know how to show that it is differentiable in a good way ...
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80 views

Let $f$ be a continous function $\mathbb R \to \mathbb R$ and $x(t)$ be of class $C^1$ given by the solution of: >$x' - f(x) = t^2$

I can't find a way to solve this one, although it seems to be quite basic: Let $f$ be a continous function $\mathbb R \to \mathbb R$ and $x(t)$ be of class $C^1$ given by the solution of: $x' ...
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49 views

Let $f(x)$ be differentiable at $\mathbb R$, s.t $|f^\prime (x)| \le 6$ . Its given that $f(1)=20$, and $f(9)=68$ , prove that $f(7)=56$.

Let $f(x)$ be differentiable ate $\mathbb R$, s.t $|f^\prime (x)| \le 6$ for every $x$ in $\mathbb R$. its given also that $f(1)=20$, and $f(9)=68$ , prove that $f(7)=56$. I'm thinking about applying ...
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44 views

Lagrangian Method for Christoffel Symbol and (non-)holonomic basis

I rencently learned about the lagrangian/variational method for computing Christoffel symbols. Let $\mathcal{M}$ be a $m-$dimensional manifold with $g_{ij}$ being the metric tensor components and ...
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1answer
66 views

How to express the second derivative of $f^{-1}$ in terms of $f'$, and $f''$?

I have this excercise: "let $y(x)$ be a function, in $x\in (a,b)$ exist the inverse function $x(y)$. If $y(x), x(y)$ have 2 derivatives in their definition domain, by $y´(x), y´´(x)$ express: ...
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92 views

Application of differentiation.

A sphere, of radium 26cm, has circular cylinder inscribed within it sycg that edges of the 2 circular ends of the cylinder are always on the surface of the sphere.At a particular instant, the radius ...
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31 views

chain rule of $\frac{d^ny(p(t))}{dt}$

In the book, "Introduction to Ordinary Differential Equation by Agarwal," it says that derivative of $$\frac{d^ny}{dt}(p(t)) = \sum_{i = 0}^{n}p_{ni}(t)\frac{d^iy}{dt}$$ where $p_{ni}(t)$ are some ...
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77 views

Word problem about finding the inverse derivative

I have the following word problem. I need to find and interpret the meaning of the inverse derivative of a function. At a gas station, the function f(p) is the number of gallons of gasoline sold when ...
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99 views

Prove that $x+\sin x$ is strictly increasing

I have a function $f(x)=x+\sin x$ and I want to prove that it is strictly increasing. A natural thing to do would be examine $f(x+\epsilon)$ for $\epsilon > 0$, and it is equal to ...
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2answers
164 views

Differentiate w.r.t a matrix

If I define $S=H+ae^T/n$, where $a_i=1$ if $h_{ij}=0$ for all $j$, $a_i=0$ otherwise and $e$ is a column of 1's. How to do the differentiation w.r.t $h_{ij}$? I know $a$ is somehow related to ...
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1answer
42 views

Having a function $f(t)$, what is $\frac{\mathrm d(\mathrm df/\mathrm dt)}{\mathrm df}$ equal to?

Having a function $f(t)$, what is $\dfrac{\mathrm d(\mathrm df/\mathrm dt)}{\mathrm df}$ equal to? I attempted to approach this using the first principles (of differentiation) and got a limit of ...
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1answer
29 views

Finding $x$ values of tangents from derivatives with literal coefficients

Given $$f(x) = ax^2 (2x-1)^{-1}=\frac{ax^2}{2x-1}$$ find the $x$ values where the tangent is horizontal. Show all steps and express the derivative in simplified form.
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33 views

Differentiation of inverse trigonometry function.

I'm given a function $$y=8\sin ^{-1}\left(\frac{x}{4}\right)-\frac{x\sqrt{16-x^2}}{2}$$ Can anyone give me some hints or guides how to differentiate it? Thanks. Because what I got from my working is ...
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2answers
79 views

How to prove that a function is decreasing?

I would like to inquire whether there is a simple way to prove that a function is decreasing or not. For example how would I prove that the function $$Y = (X^.5 - 1)/0.5$$ is decreasing? I am not ...