# Tagged Questions

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### $f(x)$ is everywhere differentiable on $[a,b]$ then give examples

$f(x)$ is everywhere differentiable on $[a,b]$ then give examples for each (they are independent) (1) $f'(x)$ is not Riemann integrable (2) $f''(x)$ does not exist (3) $f'(x)$ is not continuous
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### Show a polar function's diffrentiability

I need to show that $f(r,\theta)=r\sin(2\theta)\ r>0$ is differentiable at each point in its domain, and also decide whether it's $C^1$ or not. How should I approach this?
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### Identifying f and a when given the formula for the derivative of f?

(Only need help with b) I tried to say that $f(a+h) -f(a) = (a+h)^{10}$ but I am getting nowhere. If $f(a+h)$ for $a=1$ is $(1+h)^{10}$, then $f(a)$ would have to be $0$ but then $f(a)$ would ...
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### What do we mean by derivative of a function? What does it tell? [duplicate]

Taking the derivative of any kind of function is easy but I don't know why we take the derivative? Like $f(x)=x^2$ has the derivative $2x$, so what does it mean? I don't know how to define ...
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### can you differentiate $y(x)=x^4 - 2x^2+8x$

Can you help me differentiate $$y=x^4 -2x^2+8x$$ with respect to $y$? Thank you.
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### Differentiate Piecewise Functions

$$f(x) = \left\{\begin{array}{cl}x^3 \sin\frac{1}{x}, & x > 0\\ x \sin(x) & x \leq 0 \end{array}\right.$$ How do I find $f'(x)$? I tried using the definition of derivatives but it got me ...
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### How to determine whether a piecewise function has a derivative?

Could someone show me a worked example of showing whether a piecewise function is differentiable at some $x=a$? I can show that it is continuous at $a$, as the limit as $x\to a$ (from both sides) ...
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### Derivative $\frac{d}{dx} \ln(x+ \sqrt[]{ x^{2} + y^{2} })$

$$\frac{d}{dx} \ln(x+ \sqrt[]{ x^{2} + y^{2} })$$ What I've done so far: $$1+\frac{0.5(x^{2})^{-0.5}2x}{x+\sqrt{x^{2}+y^{2}}}$$ $$1+\frac{\frac{x}{(x^{2})^{0.5}}}{x+\sqrt{x^{2}+y^{2}}}$$ ...
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### Algebraic issues with the calculation of the second derivative of $(a+be^x)/(ae^x+b)$

I'm trying to work out the 2nd derivative of $\dfrac{a+be^x}{ae^x+b}$ I have $f''=\dfrac{(ae^x+b)^2(b^2-a^2)e^x-2ae^x(ae^x+b)(b^2-a^2)e^x}{(ae^x+b)^4}$ There are so many terms, and I'm seriously ...
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### How to find the derivative of improper integral with variable upper limit?

I have the integral from $-\infty$ to $y^2$ of the function $(e^{-|x|})$ and I need to find the derivative of this. That is, $$\frac{d}{dy} \int_{-\infty}^{y^2} e^{-|x|}\,dx$$ Usually derivative ...
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### How to find the derivative of $e(x) = \frac{x^2 + 80x + 40f}{rx}$? [closed]

Here $f$ and $r$ are constants. $$e(x) = \frac {x^2 + 80x + 40f}{rx}$$
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### Does a word problem provide all information?

A while ago I asked a similar question about word problems and assumptions. Is it a definition or an accepted-fact that word problems provide all information about the relevant existence/situation in ...
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### What is the rule behind this derivative?

$$\dfrac{\rm d}{{\rm d}t}\big(\sin^2(t)\big)=\sin(2t).$$ I don't understand what is the rule behind this derivation. I had tried to first rerivate sin() and then to derivate the square function, but ...
Let $$f(x)=\begin{cases}x^2-3, & x<0;\\-3, & x\geq 0.\end{cases}$$ (a) Find the value of $x$ where $f$ is discontinuous (b) Find the value of $x$ where $f$ is non-differentiable ...
I know this is really basic, but how do I differentiate this equation from first principles to find $\frac{dy}{dx}$: $$y = \frac{1}{x}$$ I tried this: \begin{align} f'(x) = \frac{dy}{dx} & ...