Differentiation

What is an open neighbourhood of the point a?

\left(a-\delta,a+\delta\right),\delta>0

If f is a function defined at least in an open neighbourhood of the point a, when do we say f is differentiable at a?

if $lim_{x\rightarrow a}\frac{f\left(x\right)-f\left(a\right)}{x-a}$ exists

For a function f(x), and $a\in\mathbb{R}$, what is equivalent to f is differentiable at a?

$\exists F_{a}\left(x\right)s.t.f\left(x\right)=f\left(a\right)+F_{a}\left(x\right)\left(x-a\right)$ and $F_{a}\left(x\right)$ is continuous at x=a ($f^{\prime}\left(a\right)=F_{a}\left(a\right)$)

If f is differentiable at a, what can we say?

f is continuous at a

If g(y) is differentiable at y=k and f(x) is differentiable at x=g(k), then what can we say about $fog$?

it is differentiable at y=k and $\left(fog\right)^{\prime}\left(k\right)=f^{\prime}\left(g\left(k\right)\right)g^{\prime}\left(k\right)$

Let f(x) be a strictly monotonic and continuous function on [a, b], and let g(y) be the inverse of f so that g is strictly monotonic and continuous by the Inverse Function Theorem. If f is differentiable at $l\in\left(a,b\right)$ and that the derivative $f^{\prime}\left(l\right)\ne0$, what does the Inverse Rule state?

g is differentiable at $k=f\left(l\right)$ and $g^{\prime}\left(k\right)=\frac{1}{f^{\prime}\left(l\right)}$

What is the sine-angle-tangent inequality?

• $$0\le\sin\alpha\le\alpha\le\tan\alpha,\forall\alpha\in\left(0,\frac{\pi}{2}\right)$$

• $$\left\vert\sin x\right\vert\le x,\forall x\in\mathbb{R}$$

What is the sine subtraction formula?

$$\sin y-\sin x=2\sin\left(\frac{y-x}{2}\right)\cos\left(\frac{y+x}{2}\right)$$

What is the cosine subtraction formula?

$$\cos y-\cos x=-2\sin\left(\frac{y-x}{2}\right)\sin\left(\frac{y+x}{2}\right)$$