# Tagged Questions

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

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### If $|a+b+p+q|=\frac{k}{18}$, then find the value of $k$

Let $$f(x)= \begin{cases} ax(x-1)+b & x<1 \\ x+2 & 1\leq x\leq 3 \\ px^2+qx+2 & x>3 \end{cases}$$ be continuous for all x except $x=1$ but $|f(x)|$ is ...
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### Derivative of degree k for $f(t)$ $=$ $1 \over {1 + t}$

Given $f: \Bbb R \setminus \{-1\} \rightarrow \Bbb R$, $f(t)$ $=$ $1 \over {1 + t}$, I would like to calculate the derivative of degree $k$. Approach First, we try to examine if ...
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### 2nd derivative of xy w/ respect to x?

$$\frac{d^2}{dx^2}xy$$ I know it equals zero but I don't know the in between-steps. I'm using it to prove Newtons Laws work in any frame of reference. So say two guys start from the same point and ...
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### Find number of tangent to given curve

The number of tangent to curve $x^\frac{3}{2} +y^\frac{3}{2} = a^\frac{3}{2}$ where the tangents are equally inclined to axes, is My work $$\frac{dy}{dx}=-\sqrt\frac{x}{y}$$ From above we can say ...
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### Could this be proper notation for an antiderivative? Does this notation even exist?

If we define $f(x)$ as some arbitrary function, then we can define $f'(x)$ or $f^{(1)}(x)$ as the first order and $f''(x)$ or $f^{(2)}(x)$ as the second order. My question is: Is there sure thing ...
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### How is the class related to derivability?

Good evening to everyone. I have a question where they require me to find the derivability. After I read the answer sheet I saw that the function has the class $C^1$. How is the class related to ...
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### Why $\frac{d}{dt}f(x+t(y−x))<0$ if $x < y, f(y) < f(x)$

Here excerpt from a book: Аssume that $f$ satisfies $\nabla f(x) \ge 0$ for all $x$, but is not nondecreasing, i.e., there exist $x,y$ with $x < y$ and $f(y) < f(x)$. By ...
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### $f: \Bbb R^2 \to \Bbb R$ whose partials exist. Show: $\partial _xf \:\:\mathrm{continuous} \Rightarrow f \:\:\mathrm {differentiable}$

Let $f: \Bbb R^2 \to \Bbb R$ be a function whose partial derivatives exist. Now i have to show: $$\partial _xf \:\:\mathrm{continuous} \Rightarrow f \:\:\mathrm {differentiable}$$ Any tipps on how ...
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### Using the chain rule for cos and sin functions

I am having issues with derivatives containing chain rules. I know there is multiple threads already but after reading a few, I still find myself confused. I also checked the actual answer following a ...
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### Definition of derivative to calculate $x\sqrt{|x|}$ at $x=0$

So the question says use the definition of the derivative to calculate the derivative of $x\sqrt{|x|}$ at $x=0$. I understand the definition of derivative but have no idea where to go from there to ...
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### Why is it true that $S'(t)/S(t) = d log(S(t)) / dt$?

I came across this identity in derivation of the hazard rate in survival analysis.
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### Derivative of a characteristic polynomial at an eigenvalue

Let $p(\lambda)$ be the characteristic polynomial of an $n\times n$ matrix $A$. We know that the roots of $p(\lambda)$ are the eigenvalues of $A$, hence the sum of the roots of the polynomial (taking ...
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### How would I find the nth derivative of a function, where n is imaginary? What about where n is not a constant? [duplicate]

Forgive me if this question has already been asked. I was unable to find anything relevant to this question. The $n$th derivative of a function, $f^n(x)$ is well-defined for $n\in\mathbb{Z}^+$. As ...
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### What should i conclude from the following workout?

We know that for any value of $x$ other than $0$, $a^x\ne e^x$ where $a>e, a\in R^+$ but we do know that for some value of $p$, $$pa^x=e^x\ldots(1)$$ you see $p$ is a positive number because of ...
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