# 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 the $n$-th derivative of $f(x)=e^{x}\sin(x)$

I am trying to find the general form for the $n$-th derivative of $f(x)=e^{x}\sin(x)$. I have calculated the derivatives up to $5$, but I am having trouble coming up with a general rule. Here is my ...
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### The uniqueness of solution to $1+2^{\log_3x}=x$

I have this equation: $$1+2^{\log_3x}=x \text{ where } x \in \mathbb{R}$$ Anyone can immediately see the solution, $x=3$, but the remaining problem is to prove that $x$ is the unique solution. We can ...
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### A particle moves along the x-axis find t when acceleration of the particle equals 0

A particle moves along the x-axis, its position at time t is given by $x(t)= \frac{3t}{6+8t^2}$, $t≥0$, where t is measured in seconds and x is in meters. Find time at which acceleration equals 0. ...
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### Prove that $f(z(t)), f(w(t))$ are perpendicular at $t=0$

I have the following problem but I'm not sure if my proof is correct: Let $f(z)$ be a holomorphic function. Let $z(t)=a(t)+ib(t)$ and $w(t)=c(t)+id(t)$ be perpendicular at $t=0$. We have shown in ...
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### From current plot $y=f(x)$ get plot $dx/dy$ vs $y$

I have a plot $y = f(x)$ where $y$ is voltage and $x$ is capacity. Now I want get from this graph the $dx/dy$ vs $y$ plot. How can I get this new graph?
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### What is the derivative of $\int_{-10}^{-3} e^{\tan(t)} \,dt$ with respect to x?

We were learning about the Fundamental Theorem of Calculus today in my high school and the above integral came up as an example of an integral with a "constant" value. At first I accepted that the ...
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### (beginner question) How to find points where a series stops being flat, or becomes flat?

I have a series of distributions that fall into three classes: series is flat(tish), then falls, then becomes flat(tish) again series is flat and remains flat series is flat(tish), then falls, and ...
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### Finding a two-variable function that is distinct from another on every open disk, with specifics.

Consider the two-variable function $$f(x, y) = \sin(x) + \cos(x) + y^2.$$ Find a two-variable function $g(x, y)$ that is distinct from $f(x, y)$ on every open disk which contains the point $(1, 2)$ ...
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### Derivative of the area of a circle - Unsure why my answer is incorrect

The initial radius of a circle is $3$cm, but it grows at a rate of $\frac{1\text{cm}}{\text{second}}$ The problem is taken from this Khan Academy video I work out my answer in a similar way to his ...
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### Is the following function continuously differentiable?

I am given a piecewise function, $f(x,y)=(xy,\frac{x^4}{x^2+y^2})$ if $(x,y) \neq 0$ and $f(x,y)=(0,0)$ if $(x,y)=0$. Thus $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$. I am asked if this is ...
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### $f \in C^1$ defined on a compact set $K$ is Lipschitz?
Let $f: \Omega \subseteq \mathbb{R}^N \to \mathbb{R}^M$ be $C^1$, and $K \subseteq \Omega$. Prove that $f \mid_K$ is Lipschitz. Letting $x,y \in K$, I know that $f$ is loccaly Lipschitz, I thought ...