# Tagged Questions

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

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### Integral/Derivation of Modulo/Greatest Integer

I've been trying to find the integral and derivatives of the modulo function. For those of you who are not aware, it is a primarily Computer Science based operator that is defined as the remainder ...
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### Why is it legitimate to perform multiplication with differentials dx?

Why is it legitimate to perform multiplication with differentials $dx$? For instance, from the statement $dy = 5dx$ one derives $\frac{dy}{dx} = 5$. I learned $\frac{dy}{dx}$ as a notation to mean ...
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Find a function $g(x)$ that is as simple as possible s.t. $\exp(x)-1=\mathcal{O}(g(x))$ for $x\to 0$. Claim. Such a possible function is $g(x)=x$. Proof. Using the definition of the class $\mathcal{... 2answers 33 views ### Differential Equation, linear or non-linear? I am new to the area of solving differential equations, and I came across the following differential equation and was wondering whether it was linear or non-linear:$dy/dx= x^3 + y^3$I would have ... 1answer 87 views ### Minimizing$f(x)=A^{\frac{tx-1}{x-1}} \left( c^x \frac{\Gamma(0.5+x)}{\sqrt{\pi}} \right)^{\frac{1-t}{x-1}}$subject to the constraint Let$f(r)be a function defined as follows \begin{align} f(x)=A^{\frac{tx-1}{x-1}} \left( c^x \frac{\Gamma(0.5+x)}{\sqrt{\pi}} \right)^{\frac{1-t}{x-1}} \end{align} where0 < A,c$and$ t\in (0,...
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We all know that if $z=f(x,y)$, then the total derivative of $z$ is given by the formula $\Bbb d z = \dfrac {\partial f} {\partial x} \Bbb d x + \dfrac {\partial f} {\partial y} \Bbb d y$. My ...
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### $xf'(x) = αf(x)$. How to prove that $f(x) = cx^\alpha$?

Let $f$ be a differentiable function such that $xf'(x) = \alpha f(x)$ for all $x > 0$. How do I show that $f(x) = cx^\alpha$ for some constant $c$? I have $f'(x) = \alpha f(x)/x$ , and I can see ...
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### Show there exists $\xi \in [a,b]$ such that $g(\xi)\int_a^\xi f(x)\text{d}x=f(\xi)\int_\xi^b g(x)\text{d}x$

Assume $f(x),g(x)$ is continuous on $[a,b]$. show that there exists $\xi \in [a,b]$, such that $$g(\xi)\int_a^\xi f(x)\text{d}x=f(\xi)\int_\xi^b g(x)\text{d}x$$ I tried to use intermediate value ...
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### Why Does $f(x) = x\sqrt{x+3}$ Only Have One Critical Point?

I am trying to find the critical points of the function $f(x) = x\sqrt{x+3}$, then by using the First Derivative Test, determine which ones are a local maximum, local minimum, or neither. Using the ...
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### Find the maximum area of a right triangle with a constant perimeter P.

I have been learning calculus from a tutor and I have been trying to solve a problem that he gave me. The problem is to find the maximum area of a right triangle with a constant perimeter $P$. To ...
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### Related Rates Involving a Graphical Setting

Please help me with this related rates problem: A point P is moving along the curve whose equation is $y=\sqrt{x}$. Suppose that $x$ is increasing at a rate of $4 \frac{units}{s}$ when $x=3$. How fast ...
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### Examining second derivatives.

If I have a set of continuous functions X on an interval [a,b], such that f(a)=f(b)=0 for all functions in the set. Is it possible to create a mapping from X to C[a,b] by using the second derivative ...
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### Two non-differentiable functions whose product is differentiable.

So I was wondering while studying analysis if there is any case where two functions aren't differential at $0$ (kind of like $1/x$) but is differentiable at 0 when combined (i.e. $fg$). I mean this ...
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### Implicit derivatives and logarithmic derivatives: $\left(x^{\sqrt{x}}\right)'=?$

How would I find the derivative with respect to $x$ of $$y = \left(x^{\sqrt{x}}\right)'.$$ I can find the correct answer using the method of logarithmic differentiation that my book mysteriously ...
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### Total Derivative as a Linear Map

I am currently studying Calculus on Manifolds .I am studying Spivacks book along with Munkres and J.Shurmans notes since i might not understand something from one book to another.What i noticed is so ...
### Proving that the tangent line to the graph of $f$ at $(a, 1/a^2)$ intersects $f$ at one other point
This comes from Spivak's Calculus (problem 2b from chapter 9). Here, $f(x)=1/x^2$. My approach was as follows: I took $f'$ and wrote down the tangent line to $f$ at $(a, 1/a^2)$, namely \$y_a=-2/a^...