# Tagged Questions

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

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### Finding constants with differentiation [closed]

The curve y= f(x) for which f'(x)= 4x+k, where k is a constant, has a turning point at (-2, -1). a) Find the value of k. b) Find the coordinates of the point at which the curve meets the y-axis.
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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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### total derivative of function

Suppose one has a function: $G(x,y) = H(x,y) + L(x,y)$ Is it possible to evaluate the total derivative of $G$ with respect to $H$? That is, is it possible to compute, $\frac{d G}{d H}$ ?
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### Prove the Inequality $\frac{1}{1-x}-\frac{x(3-x)(2-x)(13x^4-50x^3+89x^2-84x+36)}{4(1-x)(2x(1-x))^2}<1$

Can anyone suggest any hints to prove the following inequality: $$\frac{1}{1-x} - \frac{x(3-x)(2-x)(13x^4 - 50x^3 + 89x^2 - 84x + 36)}{4(1-x)(2x(1-x))^2} < 1,$$ for all $x \in (0,1)$?
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### How to solve this implicit differentiation problem concerning arcsin?

My overarching question is about differentiating when you have these inverse trig functions, but listed below is the specific question I am trying to solve. If you help me with the problem, it'll help ...
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### Calculus, limit at infinity exists, bounded second derivatives

Let $f:[0,\infty) \to \mathbb{R}$ twice differentiable. If $f''$ is bounded and $\lim_{x\to \infty} f(x)$ exists, show that $\lim_{x\to \infty} f'(x) = 0$. Update: So following the link from one of ...
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### Find the first four nonzero terms of the Taylor series for $\sin x$ centered at $\frac{\pi}6$

Find the first four nonzero terms of the series for $f(x)$ centered at $a$, using the definition of Taylor series. $$f(x) = \sin(x),\quad a=\pi/6$$ I got this: 1st term: $1/2$ 2nd: $\sqrt{3}/2$ ...
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### determine all (x,y) of the line Normal to an Ellipse

Hi everyone I have a question that requires me to determine the (x,y) coordinates of all points that intersects the x-axis on this ellipse when the normal line has a slope of -4, and I'm curious to ...
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### Usage of implicit function theorem for $f(x,y)=x^2+2xy-y^2-a^2$

Find the derivative of the following implicit function with the implicit function theorem: $$F(x,y)=x^2+2xy-y^2-a^2$$ My attempt for this task: $$F(x,y)=0 \Leftrightarrow (x,y)=(a,0)$$ ...
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### Derivative of $2(1-L)^{1/2} L^{1/2}$

I have never been good at math. How can i derive the top equation to get the last equation at the bottom. I've checked Wolframalpha and various other derivative calculators and they have different ...
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### Gradient of piece wise constant quantum control problem to steer system evolution to a target state

I'm looking for an exact gradient for the piece wise constant control of a quantum system to steer it towards a desired state at time T. It is worth mentioning, the Hamiltonians have been expanded ...
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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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### why do we use dy/dx as ratio though it is not while solving the problems of integration by substitution [duplicate]

According to my knowledge dy/dx is not a ratio. Then while solving the problems of integration by substitution how can we use it as ratio. Because of we have dx/dt =f(x). Then while shoving it by ...
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### Symbol of differential operator and change of coordinates

Some time ago I posted the question about the change of coordinates in differential operator. Here is the relevant discussion Symbol of differential operator transforms like a cotangent vector The ...
I have $f(x) = \begin{cases} \frac{e^x-1}{log(x+1)} & \quad \text{if } x>-1 ,&x\not=0 \\ 1 & \quad \text{if } x= 0\\ \end{cases}$ I need the tangent line of ...