# Tagged Questions

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

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### Diffrenece of exponential functions

Prove that the function $f: \Bbb R \to \Bbb R$, $f(x)=2016^x-2015^x+x$ is strictly increasing. I tried to find the derivative, but it didn't help me.
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### Multi-derivative of standard normal CDF

I am trying to solve the following $m$th-derivative of standard normal cdf, $$\frac{\text{d}^m}{\text{d}a^m}\Phi \left(\frac{a+\mu u}{\sqrt{u}}\right),$$ where $m> 0$ is an integer , $\mu>0$,...
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### How to calculate the Bouligand derivative (B-derivative)

Let $H(x)=\min (f(x),h(x))$ where $f$ and $h$ are continuously differentiable functions from $\mathbf{R}^n$ to $\mathbf{R}^1$. The Bouligand derivative (B-derivative) $BH(z)$ at $z$ of $H$ is given ...
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### Shouldn't l'Hopital's rule work for every limit, not just indeterminate forms?

Why does taking the ratio of $f'(x)$ to $g'(x)$ as $x \to a$ give you the correct limit when $f(a)$ and $g(a)$ $= 0, \infty, -\infty$ , but not for other values of $a$? If the rationale for using ...
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### Computer Vision Models 4.3 - Derivative of Summation

I am reading through the Computer Vision: Models, Learning, and Inference book to get an understanding of computer vision. The author describes the high-level steps taken to arrive at one of the ...
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Question: Find the intervals in which the following function is strictly increasing or decreasing: $(x+1)^3(x-3)^3$ The following was my differentiation: $y = (x+1)^3(x-3)^3$ $\frac1y \frac{dy}{dx} ... 2answers 21 views ### How do you check which intervals a cubic function will increase and in which intervals it will decrease? I was trying to find the intervals in which the cubic function$4x^3 -6x^2 -72x + 30$would be strictly increasing and strictly decreasing. I managed to get the fact that at the values {-2,3} the ... 3answers 46 views ### How can I differentiate$f(x,z(x,y))$w. r. to x How can I differentiate$f(x,z(x,y))$w. r. to x If$z(x,y)=c=\text{constant}$and$\hat{y}=f(x,c)$then what is$d\hat y/dx$If I just differentiate$f$w.r. to$x$without knowing whether$z$is ... 0answers 135 views ### Showing$x^*$is a saddle point Let$f \in C^2(U;\mathbb{R})$, where$U$is an open subset of a normed space$\mathbb{X}$. Let$x^* \in U$be a critical point of$f$. Suppose there exists$u^{-}$and$u^{+} \in \mathbb{X}$such that ... 2answers 40 views ### Prove that if$x = \sqrt{a^{\sin^{-1} t}}$and$y = \sqrt{a^{\cos^{-1}t}}$then$\frac{dy}{dx}$=$-\frac{y}x$Prove: If$x = \sqrt{a^{\sin^{-1} t}}$and$y = \sqrt{a^{\cos^{-1}t}}$where$\sin^{-1}$and$\cos^{-1}$are inverse trig function, show that$\frac{dy}{dx}$=$-\frac{y}x$Unfortunately I don'... 1answer 35 views ### Derivative with a “mixed” discontinuity I read that the derivative of a function can never have a "jump" discontinuity, but only essential discontinuity. My question is, can the derivative have a "half essential and half jump" discontinuity,... 1answer 50 views ### Why can't I change an equation before I differentiate it? So recently I was reviewing calculus, and I tried to differentiate the equation:$(x^2-y^2)/(x^2+y^2)=1/2$The first thing I did was make the equation easier to differentiate by multiplying the whole ... 1answer 30 views ### How to “mix” differentiation under the integral sign with the fundamental theorem of calculus? The following function appeared before me today, and I don't know how to differentiate it: $$f(t) = \int_0^t h(s,g(t,s))\,{\rm d}s.$$ Assume that all functions involved are$C^\infty$(or whatever we ... 0answers 44 views ### Parital derivative of a **scalar** loss function w.r.t. a **row vector** of a matrix Still struggling the partial derivative of a scalar function w.r.t a row vector of a matrix, what is the way to solve such question, though I learned from here about the partial derivative of a scalar ... 2answers 42 views ### Uniform limit of a sequence of bounded derivatives is a bounded derivative? Let$\{f_n\}$be a sequence of differentiable functions on$\mathbb R$such that$f_n'$is bounded for each$n$; if$\{f_n'\}$converges uniformly to$f$on$\mathbb R$then is it true that$f$is ... 1answer 54 views ### For the differentiation of$x^{\frac23} + y^{\frac23} = a^{\frac23}$, why is the substitution$x = a \cos^3\theta$legal? While looking at a solution for finding the derivative of$x^{\frac23} + y^{\frac23} = a^{\frac23}$, the book uses: Let$x = a \cos^3\theta$and$y = a\sin^3\theta$However, why would that ... 2answers 59 views ### A little advice on using the chain rule for differentiation I must be a little rusty, but how would I evaluate the following: $$\frac{d}{dr}\left(1-\frac{b(r)}{r}\right)^{-1}$$ My stickler is that$b$is a function of$r$... 1answer 36 views ### If$f$is a real valued function, complex differentiable at$z_0$, then$f'(z_0)=0$Cannot understand this proof that a real-valued function which is complex differentiable must have derivative at that point equal to zero. I just don't understand how the last statement in bold is ... 1answer 24 views ### Partial derivative of$f(x,y) = x \arctan\left[\frac{x}{y}\right]\$
Can someone help me calculating a partial derivative of the function: $$f(x,y) = \begin{cases} x \arctan\left[\frac{x}{y}\right] & \text{if } y \neq 0 \\ 0 & \text{if } y = 0 \end{cases}$$ ...