# Tagged Questions

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

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### Show differentiability of $f(x) = \sqrt{x^4 + y^4}$ in $(0,0)$

I'm trying to show the differentiability of $$f:\mathbb R^2 \to \mathbb R\text;\quad f(x) = \sqrt{x^4 + y^4}$$ in (0,0). Here's my attempt: Since $\partial_xf(x,y) = \frac{4x^3}{2\sqrt{x^4+y^4}}$ we ...
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### Antiderivative for a function for integration

I have: $$f(x)=\cos(x) \times e^{\sin(x)}$$ and the fitting Antiderivative: $$F(x)=e^{\sin(x)}$$ Can someone please explain to me how I get from $f(x)$ to $F(x)$? In small steps? Thanks!
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### Manipulation of Polynomials and Goodness-of-Fit

This question is directly connected with my problem in Python section HERE. Basically I've programmed a method with mathematical and physical background that forms a polynomial of selected degree ...
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### How to find $f(x)$ given $f'(x)$

I am dealing with a problem where I have the derivative $f'(x)$ in function of its antiderivative $f(x)$. How can I solve this? $f'(f(x))$ is linear (just assume any linear function, for sake of ...
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### When do differential operators commute?

Given that the equation of motion of a particle placed on the apex of Norton's Dome is $$\frac{d^2 r}{dt^2}=r^{1/2}\qquad\longleftarrow\text{as proved in this previous question}\tag{1}$$ Solve this ...
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### Find the number of roots lie in interval

Let $a \in R$ and let $f : R \rightarrow R$ be given by $f(x)=x^5 -5x + a$ Then $f(x)$ has three real roots if $a \gt 4$ $f(x)$ has only one real roots if $a \gt 4$ $f(x)$ has ...
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Can anyone give a rigorous proof of the derivative of this type of function? Specifically showing, $\frac{d(x^w)}{dx} = wx^{w-1}$ for a real $w$? I tried to use the Taylor series expansion for (x+... 1answer 45 views ### Sinc function derivative formula I was trying to find a formula for the derivative of the following function $$f_{\alpha}(x) = \frac{\sin(\alpha x)}{x}$$ Since $$\sin^{(k)}(\alpha x) = \alpha^k g_k(\alpha x),$$ where $$g_k(\... 2answers 43 views ### Prove or Disprove that f(x) cannot be C_3 Consider a function y=f(x) with a single argument x with the real number line as its domain. Fix a real number Q>0, and suppose the following: f(x)=f'(x)=f''(x)=0 for all x from the interval ... 3answers 37 views ### How to show this function is differentiable everywhere? Let f: \mathbb{R} \rightarrow \mathbb{R}: x \mapsto x |x|. I plotted this function with Wolfram, and I see it is smooth everywhere so I figure it must be everywhere differentiable. I wanted to ... 2answers 49 views ### Show that the equation of motion for a particle on Norton's Dome is \frac{d^2 r}{dt^2}=r^{1/2} A particle sits at the top of a dome, whose height drops away from the centre, with a drop given by$$h=\frac{2r^{3/2}}{3g}$$where g is the acceleration due to gravity, and r is a coordinate ... 1answer 24 views ### Modulus in Calculus? Does Modulus function has any effect during differentiation and integration of a quantity? For example: Let two velocities be:$$ v_1= (t-2) m/s $$and$$ v_2=|t-2|m/s $$If we differentiate them ... 1answer 99 views ### Proving a function is not differentiable, when its partials are not continuous Let$$f(x,y)=\frac{y \sin (3 x)}{\sqrt{x^2+y^2}},$$and f(0,0)=0. I'm trying to prove that it's not differentiable in (0,0). Some my plan was to compute the limit of the definition of ... 1answer 34 views ### Derivative for function at particular value How to find \dfrac{\text{d}y}{\text{d}x} at x=\dfrac{\pi}{4} for:$$y=(\log_{\cos}\sin(x))(\log_{\sin}\cos(x))^{-1}+\sin^{-1}\left(\frac{2x}{1+x^2}\right)$$6answers 61 views ### Solving second order differential equation 3 How do one solve this?$$\begin{align}\frac{d^2y}{dt^2}+e\frac{dy}{dt}+1=0,&&y(0)=0,&&\frac{dy(0)}{dt}=1\end{align}$$The exact solution of above equation is$$y(t)=\frac{1+e}{e^2}(1-... 1answer 35 views ### Slope of this tangent Okay I have to find the slope of this tangent to the curvey=\int_0^x \frac{dx}{1+x^3}$at the point where$x=1$. My try- I integrated the expression and differentiated it afterwards to get the ... 0answers 41 views ### Let$f$be a function$f:\mathbb{R}\rightarrow \mathbb{R}$such that$f(x)f''(x)\leq 0$for all$x\in \mathbb{R}$. [duplicate] Let$f$be a function$f:\mathbb{R}\rightarrow \mathbb{R}$such that the derivative$f'(x)$and all its successive derivatives exist everywhere and also$f(x)f''(x)\leq 0$for all$x\in \mathbb{R}$. ... 0answers 43 views ### Find the derivative of$y = \frac{\sqrt[3]{4x^3+8}}{(x+2)^5}$[duplicate] Can anyone solve 1b? Find the derivatives of each of these functions. b.$y = \frac{\sqrt[3]{4x^3+8}}{(x+2)^5}$I am really confused, I used the quotient rule for differentiation of the ... 4answers 84 views ### Find the derivative of$y= \frac{(4x^3 +8)^{\frac{1}{3}}}{(x+2)^5}$[closed] How can we find the derivative of$y= \frac{(4x^3 +8)^{\frac{1}{3}}}{(x+2)^5}$? so far this is what I have done, and am confused about what to do after? sorry for the messy handwriting 3answers 142 views ### Find range of$f(x)=3^x+5^x-8^x$Find range of$f(x)=3^x+5^x-8^x$. My attempt: On observation one sees that$f(1)=0$. On taking$g(x)=\left(\frac{3}{8}\right)^x+\left(\frac{5}{8}\right)^x-1$and then observing that$g'(x)=\left(\...
Let $f:R\rightarrow R$ be a function such that all its successive derivatives exist in all $R$ and also $f(x)f''(x)\leq 0$ everywhere. If $\alpha$ and $\beta$ be two successive roots of $f(x)=0$. ...