Limits and Standard derivatives

2 Properties of limits

\(\lim_{x \to a} [cf(x)]=c\lim_{x \to a} f(x)\)
\(\lim_{x \to a} [f(x) \pm g(x)]=\lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)\)
\(\lim_{x \to a} [f(x) g(x)]=\lim_{x \to a} f(x) \lim_{x \to a} g(x)\)
\(\lim_{x \to a} \frac{f(x)}{g(x)}=\frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}\) if \(\lim_{x \to a} g(x) \neq 0\)
\(\lim_{x \to a} [f(x)]^n=[\lim_{x \to a} f(x)]^n\)
\(\lim_{x \to a} [\sqrt[n]{f(x)}]=\sqrt[n]{\lim_{x \to a} f(x)}\)
\(\lim_{x \to a} c=c\)
\(\lim_{x \to 0} \sin x=0\)
\(\lim_{x \to 0} \cos x=1\)
\(\lim_{x \to 0} \frac{\sin x}{x}=1\)
\(\lim_{x \to 0} \frac{\tan x}{x}=1\)
\(\lim_{x \to 0} \frac{\log (1+x)}{x}=1\)
\(\lim_{x \to 0} e^x =1\)
\(\lim_{x \to 0} \frac{e^x-1}{x}=1\)
\(\lim_{x \to 0} \frac{a^x - 1}{x}=\log_e a\)
\(\lim_{x \to \infty} (1+\frac{1}{x})^x=e\)
\(\lim_{x \to 0} (1+x)^{1/x}=e\)
\(\lim_{x \to \infty} (1+\frac{a}{x})^x=e^a\)
\(\lim_{x \to \infty} a^x = \infty\), when \(a>1\)
\(\lim_{x \to \infty} a^x = 0\), when \(a<1\)
\(\lim_{x \to a} \frac{x^n - a^n}{x-a}=na^{n-1}\)
\(\lim_{x \to 0} \frac{(1+x)^n-1}{x}=n\)
\(\lim_{n \to \infty} nx^n=0\), when \(|x|<1\)
\(\lim_{n \to \infty} \frac{x^n}{n!}=0\)
\(\lim_{n \to \infty} \frac{m(m-1)(m-2)\cdots(m-n+1)}{n!}x^n=0\), when \(|x|<1\)

3 Standard derivative rules and formulas

These formulas should be remembered for solving further problems:

\((cf)'=cf'(x)\)
\(\frac{d}{dx}(c)=0\)
\((f \pm g)'=f'(x) \pm g'(x)\)
\(\frac{d}{dx}(x^n)=nx^{n-1}\)
\((f g)'=f'g + fg'\) \(\Longrightarrow\) Product rule
\(\frac{d}{dx}(f(g(x)))=f'(g(x))g'(x)\) \(\Longrightarrow\) Chain rule
\((\frac{f}{g})'=\frac{f'g-fg'}{g^2}\) \(\Longrightarrow\) Quotient rule
\(\frac{d}{dx} (x) = 1\)
\(\frac{d}{dx} (\sin x)= \cos x\)
\(\frac{d}{dx} (\cos x) =-\sin x\)
\(\frac{d}{dx} (\tan x) =\sec^2 x\)
\(\frac{d}{dx} (\csc x) =-\csc x \cot x\)
\(\frac{d}{dx} (\sec x) =\sec x \tan x\)
\(\frac{d}{dx} (\cot x) =-\csc^2 x\)
\(\frac{d}{dx} (a^x) = a^x \log (a)\)
\(\frac{d}{dx} (e^x) = e^x\)
\(\frac{d}{dx} (\ln x) = \frac{1}{x}, x>0\)
\(\frac{d}{dx} (\ln |x|) = \frac{1}{x}, x\neq 0\)
\(\frac{d}{dx} (\sin^{-1}x) = \frac{1}{\sqrt{1-x^2}}\)
\(\frac{d}{dx} (\cos^{-1}x) = -\frac{1}{\sqrt{1-x^2}}\)
\(\frac{d}{dx} (\tan^{-1}x) = \frac{1}{1+x^2}\)
\(\frac{d}{dx} (\csc^{-1}x) = -\frac{1}{|x|\sqrt{x^2-1}}\)
\(\frac{d}{dx} (\sec^{-1}x) = \frac{1}{|x|\sqrt{x^2-1}}\)
\(\frac{d}{dx} (\cot^{-1}x) = -\frac{1}{1+x^2}\)
\(\frac{d}{dx} (\sin \text{h}x) = \cos \text{h}x\)
\(\frac{d}{dx} (\cos \text{h}x) = \sin \text{h}x\)
\(\frac{d}{dx} (\tan \text{h}x) = \sec\text{h}^2x\)
\(\frac{d}{dx} (\csc \text{h}x) =-\csc \text{h}x \cot \text{h}x\)
\(\frac{d}{dx} (\sec \text{h}x) =-\sec \text{h}x \tan \text{h}x\)
\(\frac{d}{dx} (\cot \text{h}x) =-\csc \text{h}^2x\)
\(\frac{d}{dx} (\log_a(x)) =\frac{1}{x\ln a}, x>0\)

4 Leibnitz’s theorem

Useful for finding \(n^\text{th}\) derivatives of product functions:

If \(y = uv\) where \(u\) and \(v\) are functions of \(x\) possessing \(\text{n}^{th}\) derivatives, then

\[\begin{equation*} \begin{split} y_n &= {}^{n}C_{0}u_nv + {}^{n}C_{1}u_{n-1}v_1 + {}^{n}C_{2}u_{n-2}v_2 + {}^{n}C_{3}u_{n-3}v_3 + \ldots + {}^{n}C_{r}u_{n-r}v_r + \ldots + {}^{n}C_{n}uv_n\\ &= u_nv + {}^{n}C_{1}u_{n-1}v_1 + {}^{n}C_{2}u_{n-2}v_2 + {}^{n}C_{3}u_{n-3}v_3 + \ldots + {}^{n}C_{r}u_{n-r}v_r + \ldots + uv_n\\ (uv)_{(n)} (x) &= \sum_{k=0}^{n} {}^{n}C_{k} u_{(n-k)}(x) v_{(k)} (x), \end{split} \end{equation*}\]

where \({}^{n}C_{k} = \frac{n!}{k! (n-k)!}\) is the binomial coefficient, \(u_{(0)}(x) = u(x)\) and \(v_{(0)}(x) = v(x)\).