Intereting Posts

Limit of $(1+ x/n)^n$ when $n$ tends to infinity
Strongly complete profinite group
Showing an ideal is prime in polynomial ring
Recognizing when a tower of Galois extensions gives a Galois extension
Must $f$ be measurable if each $f^{-1}(c)$ is?
Facts about induced representations
Looking for a function $f$ that is $n$-differentiable, but $f^{(n)}$ is not continuous
Question about the proof of the Brownian motion being a Strong Markov process from Schilling.
Simplify $2^{(n-1)} + 2^{(n-2)} + … + 2 + 1$
Representability as a Sum of Three Positive Squares or Non-negative Triangular Numbers
Evaluating the reception of (epsilon, delta) definitions
Are there any algorithms or methods to compute Landau function $g(n)$?
Show that if $c_1, c_2, \ldots, c_{\phi(m)}$ is a reduced residue system modulo m, $m \neq 2$ then $c_1 + \cdots+ c_{\phi(m)} \equiv 0 \pmod{m}$
Motivation behind the definition of tangent vectors
What is the infinite product of (primes^2+1)/(primes^2-1)?

I have just learnt about the chain rule but my book doesn’t mention a proof on it. I tried to write a proof myself but can’t write it. So can someone please tell me about the proof for the chain rule in elementary terms because I have just started learning calculus.

- Evaluating $\int_{1}^{\infty}\exp(-(x(2n-x)/b)^2)\,\mathrm dx$
- How to prove that $\lim\limits_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0\text { ? }$
- $\int_{-\infty}^{\infty}{e^x+1\over (e^x-x+1)^2+\pi^2}\mathrm dx=\int_{-\infty}^{\infty}{e^x+1\over (e^x+x+1)^2+\pi^2}\mathrm dx=1$
- Differentiating both sides of an equation
- How do we calculate the area of a region bounded by four different curves?
- What is Jacobian Matrix?
- Find Minimum value of $P=\frac{1}{1+2x}+\frac{1}{1+2y}+\frac{3-2xy}{5-x^2-y^2}$
- Definite integral over a simplex
- For every continuous function $f:\to$ there exists $y\in $ such that $f(y)=y$
- Why isn't $f(x) = x\cos\frac{\pi}{x}$ differentiable at $x=0$, and how do we foresee it?

Assuming everything behaves nicely ($f$ and $g$ can be differentiated, and $g(x)$ is different from $g(a)$ when $x$ and $a$ are close), the derivative of $f(g(x))$ at the point $x = a$ is given by

$$

\lim_{x \to a}\frac{f(g(x)) – f(g(a))}{x-a}\\ = \lim_{x\to a}\frac{f(g(x)) – f(g(a))}{g(x) – g(a)}\cdot \frac{g(x) – g(a)}{x-a}

$$

where the second line becomes $f'(g(a))\cdot g'(a)$, by definition of derivative.

One approach is to use the fact the “differentiability” is equivalent to “approximate linearity”, in the sense that if $f$ is defined in some neighborhood of $a$, then

$$

f'(a) = \lim_{h \to 0} \frac{f(a + h) – f(a)}{h}\quad\text{exists}

$$

if and only if

$$

f(a + h) = f(a) + f'(a) h + o(h)\quad\text{at $a$ (i.e., “for small $h$”).}

\tag{1}

$$

(As usual, “$o(h)$” denotes a function satisfying $o(h)/h \to 0$ as $h \to 0$.)

If $f$ is differentiable at $a$ and $g$ is differentiable at $b = f(a)$, and if we write $b + k = y = f(x) = f(a + h)$, then

$$

k = y – b = f(a + h) – f(a) = f'(a) h + o(h),

$$

so $o(k) = o(h)$, i.e., any quantity negligible compared to $k$ is negligible compared to $h$. Now we simply compose the linear approximations of $g$ and $f$:

\begin{align*}

f(a + h) &= f(a) + f'(a) h + o(h), \\

g(b + k) &= g(b) + g'(b) k + o(k), \\

(g \circ f)(a + h)

&= (g \circ f)(a) + g’\bigl(f(a)\bigr)\bigl[f'(a) h + o(h)\bigr] + o(k) \\

&= (g \circ f)(a) + \bigl[g’\bigl(f(a)\bigr) f'(a)\bigr] h + o(h).

\end{align*}

Since the right-hand side has the form of a linear approximation, (1) implies that $(g \circ f)'(a)$ exists, and is equal to the coefficient of $h$, i.e.,

$$

(g \circ f)'(a) = g’\bigl(f(a)\bigr) f'(a).

$$

One nice feature of this argument is that it generalizes with almost no modifications to vector-valued functions of several variables.

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- Number of primitive characters modulo $m$.
- filling an occluded plane with the smallest number of rectangles
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- Some question on exactness of sheaves
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- In a ring with no zero-divisors, for $(m,n) =1$, $a^m = b^m$ and $a^n = b^n$ $\iff a =b$