# Tagged Questions

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

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### Find the maximum value of the function

So I was just messing around with finding the maximum and minimum values of functions, and I came across this: $$\text{Find the maximum value of} \,\, f(x)=\frac1{x^{2x^2}}.$$ Any ideas?
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### Differentiability of function for $\Bbb{Q}$ and $\Bbb{R}\setminus \Bbb{Q}$

A function $f:\Bbb{R}\to\Bbb{R}$ is defined by $f(x)=x$, if $x$ is rational; $\sin(x)$ if $x$ is irrational. Show that $f$ is differentiable at $0$ and $f'(0)=1$. Here I'm thinking to apply ...
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### $f(x, y) = \prod_{i = 1}^n (1 + xy_i)$, what is ${{{\partial f}\over{\partial x}}\over f}$, geometric series?

Let$$f(x, y) = \prod_{i = 1}^n (1 + xy_i).$$What is$${{{\partial f}\over{\partial x}}\over f}?$$What happens when we use the geometric series?
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### Chain Rule and Vector valued functions?

Let $f: R^n \to R$ be given by $f(x) = \frac{||x||^4} {1 + ||x||^2}$ . Use the chain rule to show that $f$ is differentiable at each $x \in R^n$ and compute $Df(x)$. This vector valued stuff just ...
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### Convergence of a function of two variables

The following question has been posed to me by a student in an analysis class. For which real numbers $\alpha \gt 0$ is the function $f : \Bbb R^2 \to \Bbb R$ given by $f(x, y) = (x^2 + y^2)^\alpha$ ...
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### Prove that there exists some real number θ satisfying 0 < θ < 1 for which f '''(θ) = 0

Let f: D → R be a 3-times differentiable function defined over an open interval D, where 0 ∈ D and 1 ∈ D. Suppose that f(0) = f '(0) = 0 and f(1) = f '(1) = 0. Prove that there exists some real ...
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### Question about applying the Chain Rule with multiple variables

Let $z = u(x,y)$ and $y = y(x)$ and $u(x,y(x))$ = 0. What is the second derivative of the function $y(x)$? I tried to use chain rule but I keep making mistakes
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### Why are scale factors not always unity?

A scale factor in curvilinear coordinates is defined as $$h_v \equiv \left|\frac{\partial\vec{r}}{\partial v}\right|$$ where $\vec{r}=(x,y,z)^T$ is a position vector. The partial differential can be ...
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### Implicitly finding the derivative of $f^{-1}(x)$ given $f(x)$

Can we find the derivative of the inverse of a function implicitly by finding the derivative of the original function? For example lets say I have $f(x) = e^x$ and I want to find the derivative of ...
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### Where i am going wrong in finding normal to curve?

The question is Find the perpendicular distance between the normal to the curve $$x=a\cos t+at\sin t, y=a\sin t-at\cos t$$ and the origin. Equation is given in parameterized form. My attempt ...
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### Is there a solution to this differential equation?

I am trying to find a function $y(x)$ that is a solution to $$\left(a_3 x^3+a_1 x\right) y''(x)-\left(3 a_3 x^2+2 a_1\right) y'(x)+3 a_3\, x \,y(x)=a_0 x^4+a_2$$ I tried using mathematica but it ...
### Deduce that: $\frac{d}{dx}(u^{m}v^{n})=u^{m-1}v^{n-1}[mv\frac{du}{dx}+nu\frac{dv}{dx}]$
Deduce that: $$\frac{d}{dx}(u^{m}v^{n})=u^{m-1}v^{n-1}(mv\frac{du}{dx}+nu\frac{dv}{dx})$$ When I differentiate $\frac{d}{dx}(u^{m}v^{n})$ I get: $$\frac{d}{dx}(u^{m}v^{n})=u^{m-1}v^{n-1}(mv+nu)$$ Is ...