Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).
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If $\lim_{x \to b} f'(x) = +\infty$, then (i) $\lim_{x \to b} f''(x) = +\infty$ and (ii) $\lim_{x \to b} f''(x)/f'(x) = +\infty$
$x \in [a, b]$, $f$ is $C^2$ on $[a, b]$. For (i) I thought I had it by contradiction: Suppose it's not true, so we can find $\bar{b} \in (a, b)$ such that, for $x \in [\bar{b}, b]$ and $B > 0$, $f'... real-analysis calculus limits derivatives- 26
how to prove the equation [closed]
vessel contains 2000 litres of sauce recipe which is pumped out of the vessel to be bottled at a rate of 60 litres/hr. Preservatives are added to the mixture at a rate of 80 litres/hr to which 8g of ... calculus algebra-precalculus derivatives maxima-minima- 1
Calculate the total derivative
Calculate directly the total derivative (without using partial derivatives) of the function $f(x_1,x_2)=x_1^2-10x_2$. You may wish to use the fact that $\sqrt{x^2+y^2}\geq\frac{x+y}{2}.$ $$$$ I have ... limits multivariable-calculus derivatives- 59
Find derivative of such complicated expression.
I am trying to obtain the derivative of the following expression with respect to $x$, but not getting it correctly. $$F = \biggl\{\frac{\exp(\delta){\sigma_5}}{\sqrt{\alpha^2\sigma^2_2\sigma^2_4}}\... derivatives gamma-function hypergeometric-function- 61
Simplifying equations of partial derivatives by substitution
Let $u=\ln x$, $v = \ln y$. Then what is another form of $$x^2 \left( \frac{\partial^2 f}{\partial x^2}\right) + y^2 \left(\frac{\partial^2 f}{\partial y^2}\right) + x \left( \frac{\partial f}{\... calculus derivatives partial-differential-equations partial-derivative substitution- 139
Dealing with second derivatives in error calculation
I need to find the relative error in the error of $\log P$, i.e. $\frac{\Delta(\Delta \log P)}{(\Delta \log P)}$. I need to prove that this equals $2\frac{\Delta P}{P\log P}$. I have tried so many ... derivatives physics standard-error- 711
Differentiability of $g(x,y) := f \left(\sqrt{x^{2} + y^{2}}\right)$
Suppose the function $f : \Bbb R \to \Bbb R$ is continuously differentiable and define another function $g$ as $$g(x,y) := f \left(\sqrt{x^{2} + y^{2}}\right)$$ Under what condition is $g$ ... real-analysis multivariable-calculus derivatives scalar-fields- 169
Function of two variables is differentiable
A function $f: \mathbb{R}^2 \to \mathbb{R}$ is said to be differentiable at $(x_0,y_0) \in \mathbb{R}$ if there exists a linear transformation $\lambda$ so that $$\displaystyle\lim_{\xi \to 0} \frac{\|... real-analysis derivatives solution-verification- 1,149
Calculate total derivative directly.
Calculate directly (not via partial differentiation) the total derivative of the function $f(x_1,x_2)=x_1^2-10x_2.$ You may wish to use the fact that $\sqrt{x^2+y^2}\geq\frac{x+y}{2}.$ $$$$ For the ... limits multivariable-calculus derivatives multivalued-functions- 59
Chain Rule for 3 variables - Second order Derivative
I have a diff equation in the form of $\ddot{f}$ and $\dot{f}$ and I want to turn these into $f''$ and $f'$. Prime denotes, $' \equiv d/dy$ and dot denotes, $\dot{} \equiv d/dt$. It's given that $$\... calculus ordinary-differential-equations derivatives- 420
Proving that if derivative of f(x) = a with a>0, f(x) must go to infinity
I am interested in proving that if derivative of f(x) is a real number c, c>0, as x goes to infinity, f(x) itself must go to infinity. Seems like a common-sense statement, but don't know how I can ... real-analysis limits derivatives- 13
The derivative of sum of vector norms
This is kind of complicated for me so I have to call for your help. Let $x,y$ be vectors of size $2\times N$, and $A,B$ matrices of size $2 \times 2$. Then let $f$ be a function of scalar $a$: $f(a) = ... derivatives vectors normed-spaces- 35
Product notation in partial differentiation
For a function $f:\mathbb{R}^n\to \mathbb{R}$, is it correct to write, for any $n\in \mathbb{N}$, the expression $$ \frac{\partial^n f}{\partial x_1\cdots \partial x_n}=\frac{\partial^n f}{\prod_{i=1}^... functions derivatives notation partial-derivative products- 3,418
Is there a darboux fuction that doesn't have a primitive?
We know that if f is a differentiable fuction, then f' is a Darboux fuction due to the Darboux theorem. However, the Darboux theorem isn't know as the "characterization of fuctions with primitive ... calculus integration derivatives- 11
How can I prove this partial derivative question? [closed]
Let $z=y\ln(x^2-y^2)$. Prove that $\dfrac1x \dfrac{\partial z}{\partial x}+ \dfrac1y \dfrac{\partial z}{\partial y} = \dfrac{z}{y^2}$. calculus derivatives partial-derivative- 9
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