Tagged Questions

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

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Mathematics Colleges [on hold]

Can anyone please give me the names of US colleges with the best mathematics program for undergraduates? I came to know about Carneige Mellon university, but it does not offer financial aid to ...
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Confusion regarding uniform continuity

I was trying to check the validity of the following: If $f:\mathbb R\rightarrow\mathbb R$ and its derivative $f'$ are unbounded, then $f$ is not uniformly continuous on $\mathbb R$. To me,the ...
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How to compute a Slope for a 3 or multi dimensional equation.

If I have an equation Z=X^2+Y^2+3X+6Y+5 and want to find the slope at the point x=2 , y=1 .How do we compute it ?I know for a two dimensional equation we can compute it by differentiation of Y with ...
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Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$

Problem: If $f(x)$ is continous at $x=0$, and $\lim\limits_{x\to 0} \dfrac{f(ax)-f(x)}{x}=b$, $a, b$ are constants and $|a|>1$, prove that $f'(0)$ exists and $f'(0)=\dfrac{b}{a-1}$. This ...
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Derivative of quaternions

I am trying to calculate the Jacobian of a function that has quaternions and 3D points in it. I refer to quaternions as $q$ and 3D points as $p$ $$h_1(q)=A C(q)p$$ $$h_1(q)=q_1\otimes q \otimes q_2$$...
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Differentiate equation with parenthesis

I have a problem. I'm studying calculus, but I don't have a good math background, so I have a problem: I don't know well how to differentiate an equation with parenthesis. The equation is the ...
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Directional derivative of a differentable function

Is it always true, for a differentiable function $F:\mathbb{R}^N\rightarrow\mathbb{R}^M$, that its directional derivative along a direction $v\in\mathbb{R}^N$ is equal to the product $J_f\cdot v$, ...
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What is the way to show the following derivative problem?

If $f$ is function twice differentiable with $|f''(x)|<1, x\in [0,1]$ and $f(0)=f(1)$, then $|f'(x)|<1$ for all $x\in [0,1]$ I have tried with Rolle's theorem, but fail
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N-order differential equations

Suppose that we have n-order differential equation like $$h(x)=?$$ Is it possible to find a general solution for all n? $$(x^n+1).|h'(x)|^n=const.$$.
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How to differentiate $\ln(a^x)$?

Can someone give me the process to differentiate this (with respect to $x$)? $$\ln(a^x)$$
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How to differentiate x^(1/x)?

How to differentiate the following? $$x^{\frac{1}{x}}$$ (I know the answer is $\frac{1-\ln(x)}{x^{2-\frac{1}{x}}}$, but I do not understand how to get there) Attempt at solution I believe the ...
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Fixed point, bounded derivative

Let $p\in\mathbb{N}$. Let $f:I\to\mathbb{R}$ differentiable in the closed interval $I$ (bounded or not), with $f(I) \subset I$, and let $g = f\circ f\circ \cdots \circ f = f^p$, where $\circ$ means ...
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General chain rule help/ derivatives help.

I've been thinking too much about the chain rule and I've got myself in a muddle: Suppose $y=f(g(x))$, we can easily show that $\frac {dy}{dx} = f'(g(x))\cdot g'(x)$. I would ask please that ...
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Differentiation under the integral sign in $R^3$
Considering the heat equation, $$\frac{du}{dt}=\frac{d^2u}{dx^2}$$ if $$u(x,t)=t^{\alpha}\phi(\xi)$$ with \xi=x/\sqrt{t} \enspace then \enspace \phi \enspace satisfies \enspace \alpha\phi-(1/2)\xi\...