Tagged Questions

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

2answers
28 views

Derivative of function defined by integral of different variable

I have the following exercise which I certainly have gotten no clue about it. Let F(t) be defined: $F(t) = \int_{tan(t)}^{\sqrt{t^2+1}} e^{-tx^2}dx$ What is $F'(0)$? I have no clue about ...
0answers
20 views

Sign function identity proof

The signum function is defined by$$sgn(t)=\left\{\begin{matrix}-1, \ t<0\\0, \ t=0 \\ 1, \ t>0 \end{matrix}\right.$$has derivative$$\frac{d}{dt} sign(t) = 2 \delta(t)$$Use this result to show ...
0answers
32 views

Specific fucnction has 11 different zeros

Let $f : \mathbb{C} \to \mathbb{C}$ be given by $$f(z) = z^{11} + 4 e^{z + 1} - 2$$ Show that $f$ has 11 different zeros in the annulus $\{z \in \mathbb{C} : 1 < |z| < 3\}$. This is an old ...
3answers
34 views

Third derivative of $y=at^2+2bt+c$ and $t=ax^2+bx+c$

If $y=at^2+2bt+c$ and $t=ax^2+bx+c$. Then find $$\frac{d^3y}{dx^3}$$ Now $\frac{dy}{dx}=(2at+2b).(2ax+2b)$ but to proceed further as $\frac{dy}{dx}$ is function of $x,t$
0answers
25 views

Successive differentiation

Find the value of $y_n$ for $x = 0$,when $$y = e^{(a sin^{-1}(x))}$$. In the my book its already solved the problem is that I don't understand after a certain point ,the steps. After solving we ...
2answers
70 views

Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $f(0)=0$ and $|f'(x)|\leq1 \forall x\in\mathbb R$

Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $f(0)=0$ and $|f'(x)|\leq1 \forall x\in\mathbb R$. Then there exists $C$ in $\mathbb R$ such that $|f(x)|\leq C \sqrt |x|$ ...
1answer
31 views

1answer
30 views

Is $f'(a)\ge0$ or $f'(a)>0$

If $f$ is smooth s.t. $f<0$ on $(0,a)$ and $f>0$ on $(a,1)$ is then $f'(a)\ge0$ or $f'(a)>0$ ? Is it possible that $f'(a)=0$, maybe you have an example ?
1answer
23 views

Polynomials bounded on integers

Let $p:\mathbb{R}\rightarrow \mathbb{R}$ be a real valued polynomial, such that for all integers $0\leq i\leq n$ we have $b_{1}\leq p(i)\leq b_{2}$. Let $k=\max_{0\leq x\leq n}|p'(x)|.$ Then for all ...
1answer
31 views

Can we say that, there is a neighborhood of $x_0$ such that, $f$ is differentiable in all points of neighborhood?

Let $f:\mathbb{R} \to \mathbb{R}$, and $f$ is differentiable in $x_0$. Can we say that, there is a neighborhood of $x_0$ such that, $f$ is differentiable in all points of this neighborhood? Which ...
3answers
44 views

Find the derivative of the function when given an exponential function

$y=5{x^2}e^{3x}$ Would the rule that I use for this problem be $\frac{d}{dx} e^x=e^x$ We just started learning derivatives of exponential functions and I am a little confused on where to start with ...
1answer
72 views

3answers
95 views

Derivation of $1 = x^2+y^2$ with respect to time [duplicate]

I am studying differential algebraic equations. Given the following equation: $1 = x^2+y^2$ Differentiate this equation with respect to time. The correct solution is: $0=2x \dot x + 2y \dot y$ ...
0answers
14 views

1answer
28 views

How to prove the limit formula of the second order partial derivative?

Consider following limit formula of the second order partial derivative of a function $f(x,y,z)$: \frac{\partial^2f(x,y,z)}{\partial x^2}=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x,y,z)-2f(x,...
1answer
214 views

Are there parts of Integral Calculus that just *have* to be memorized?

Note : In this question I speak more from a calculation/operational point of view, as opposed to a more theoretical (Analysis) point of view. When studying Differential Calculus, I found that ...