Questions tagged [derivatives]
Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).
34,142 questions
3
votes
2
answers
111
views
How to solve the differential equation $x\,\mathrm dy-y\,\mathrm dx=0$?
To solve $x\,\mathrm dy=y\,\mathrm dx$,
I separated $x$ and $y$ to get
\begin{align}
\frac{1}{x}\,\mathrm dx &= \frac{1}{y}\,\mathrm dy \\
\int \frac{1}{x}\,\mathrm dx &= \int\frac{1}{y}\,\...
- 31
6
votes
0
answers
68
views
Critical points of a cubic function with two unit-circle roots
How can one prove that
$$
\{z\in\mathbb C:|z|\le1,\ |z-\tfrac23|\ge\tfrac13\}
$$
is the set of all possible critical points of the cubic polynomial
$$
p(z) = (z-1)(z-r_1)(z-r_2),
$$
where $r_1,r_2$ ...
- 6,453
4
votes
5
answers
1k
views
Prove that the Euclidean norm is not differentiable at the origin
Let the scalar field $f : {\Bbb R}^2 \to {\Bbb R}$ be defined by $f(x,y) := \sqrt{x^2 + y^2}$. Prove that the partial derivatives of $f$ at $(0,0)$ do not exist and, thus, the gradient $\nabla f (0,0)$...
- 97
2
votes
0
answers
86
views
Why can’t the derivative proof for $\sum (i+1)(i+2)x^i$ be applied directly to a specific $x$?
I was solving a infinite series problem. Then, I came across this sub-problem.
$\displaystyle \sum_{i=0}^{\infty} \frac{(i+1)(i+2)(-1)^i}{2^i}$. The trick was to set $x=-1/2$. Then, it will simply to $...
- 887
2
votes
1
answer
120
views
derivative of integral of a function with respect to the same function
While doing some calculation I came across the following term [$G=G(x,x'),$ $x$ is independent of $x'$] $$K=\frac{\partial G(x,x')}{\partial (\frac{\partial G}{\partial x'})}$$ I tried to think of it ...
1
vote
1
answer
106
views
Treating derivative like a number [duplicate]
This question stems from this one: A very weird integral equation
In the fourth step op uses the differential operator like a constant number and factors $f(x)$ out of the equation
$\frac{df}{dx} -f(x)...
5
votes
1
answer
118
views
Proving that the Distributional derivative well defined
Let be $\Omega=(-1,1)\setminus \{0\}$. Consider the function
$T=\frac{d}{dx}(\frac{1}{x})$ (Distributional derivative).
What is neccesary to proof to verify that T is well defined. I know that $\frac{...
- 199
5
votes
2
answers
164
views
Missing function in composition of two functions
Let $(f\circ g)(x) =x^4+2x^3-3x^2-4x+6$ and $g(x)=x^2+x-1$. Find $f(x)$, it seem to be $f$ will have the formula $f(x)=ax^2+bx+c$. Plugging $g(x)$ in $f(x)$, we get $$ f(x^2+x-1)=a(x^2+x-1)^2+b(x^2+x-...
- 3,292
0
votes
1
answer
50
views
Applying Leibniz's Rule to Double Integrals with Variable Limits
Consider the following double integrals:
$$G_1(z_1, z_2) = \int^{z_1}_{0} \int_{0}^{z_2 + \frac{\alpha_1}{\alpha_2}(z_1 - x_1)} \varphi(x_1, x_2) \, dx_2 \, dx_1$$
$$G_2(z_1, z_2) = \int^{z_2}_{0} \...
- 3
1
vote
1
answer
90
views
Richardson Extrapolation to the highest possible accuracy to evaluate a function derivative
I am given f(x)=cosh(x) and asked to estimate f’(0.6) using Richardson Extrapolation to the highest possible accuracy. I computed the centered difference using h1=0.2 and h2=h1/2=0.1
I got
D(0.1)=0....
- 11
0
votes
1
answer
50
views
Proof of average rate of change of a function
I was learning about derivatives and I saw that the slope of the secant line between two points is the average rate of change of the function between the two points. But, the average, as we normally ...
- 41
1
vote
1
answer
109
views
derivative of $f(x,y,z)=x^y$
This is an exercise from Spivak's Calculus on manifolds. The author asks to find the derivative of the function:
$$f(x,y,z)=x^y$$
My idea was to use chain rule to simplify $f$ to something simpler, e....
- 467
-2
votes
1
answer
65
views
Differentiable functions
If we have a function $g(x)$ defined by $g(x) = f_1(x)f_2(x)$ where $f_1(x)$ and $f_2(x)$ are non-differentiable at some points, can $g(x)$ ever be differentiable everywhere? Intuitively using product ...
-1
votes
1
answer
49
views
How to prove that $C^k(\mathbb{R})$ is a subspace of $F(\mathbb{R};\mathbb{R})$?
I'm starting my linear algebra studies and came across the following statemtent:
$E = F(\mathbb{R};\mathbb{R})$ is the vector space of the one variable real functions $f:\mathbb{R} \rightarrow \...
0
votes
1
answer
84
views
Question on derivative of $f(x,y)=\sin(xy^2)$
This is an example from Calculus on Manifolds by Spivak. After proving few theorems the author states the following example on how to differentiate some function. The author defines "helper" ...
- 467
