Calculus part 2
More about \(e\)¶
\[ e = (1 + dx)^{\frac{1}{dx}} = \lim_{\Delta x \to 0} (1 + \Delta x)^{\frac{1}{\Delta x}} = \lim_{N \to \infty} (1 + \frac{1}{N})^N \]
\[ e^x = \lim_{N \to \infty} (1 + \frac{1}{N})^{Nx} = \lim_{N \to \infty} (1 + \frac{x}{Nx})^{Nx} \]
\[ Nx \to \infty \]
\[ N \to \infty \]
\[ e^x = \lim_{N \to \infty} (1 + \frac{x}{N})^{N} \]
\[ e^x = (1 + x dx)^{\frac{1}{dx}} \]
you will see why this is usefull here
\[ \sum\limits_{n=0}^{\infty}C_n x^n = e^x \]
\[ \sum\limits_{n=0}^{\infty}C_n n x^{n - 1} = e^x = \sum\limits_{n=0}^{\infty}C_n x^n \]
\[ \sum\limits_{n=0}^{\infty}C_n n x^{n - 1} = \sum\limits_{n=-1}^{\infty}C_{n + 1} (n + 1) x^n = c_0 0 x^{-1} + \sum\limits_{n=0}^{\infty}C_{n + 1} (n + 1) x^n = \sum\limits_{n=0}^{\infty}C_n x^n \]
More about \(e\) to the \(x\)¶
\[ C_{n + 1} (n + 1) = C_n \]
\[ C_{n + 1} = \frac{C_n}{n + 1} \]
\[ C_n = \frac{C_{n - 1}}{n} \]
\[ e^0 = 1 \]
\[ C_0 = 1 \]
\[ C_n = \frac{1}{n!} \]
\[ e^x = \sum\limits_{n=0}^{\infty}\frac{x^n}{n!} \]
\[ e = \sum\limits_{n=0}^{\infty}\frac{1}{n!} \]
\[ \sum\limits_{n=0}^{\infty}C_n x^n = e^x \]
\[ \sum\limits_{n=0}^{\infty}C_n n x^{n - 1} = e^x = \sum\limits_{n=0}^{\infty}C_n x^n \]
\[ \sum\limits_{n=0}^{\infty}C_n n x^{n - 1} = \sum\limits_{n=-1}^{\infty}C_{n + 1} (n + 1) x^n = c_0 0 x^{-1} + \sum\limits_{n=0}^{\infty}C_{n + 1} (n + 1) x^n = \sum\limits_{n=0}^{\infty}C_n x^n \]
\[ C_{n + 1} (n + 1) = C_n \]
\[ C_{n + 1} = \frac{C_n}{n + 1} \]
\[ C_n = \frac{C_{n - 1}}{n} \]
\[ e^0 = 1 \]
\[ C_0 = 1 \]
\[ C_n = \frac{1}{n!} \]
\[ e^x = \sum\limits_{n=0}^{\infty}\frac{x^n}{n!} \]
quoteint rule¶
\[ (\frac{f}{g})\prime = ? \]
\[ \frac{f(x)}{g(x)} = h(x) \]
\[ g(x) h(x) = f(x) \]
\[ (g(x) h(x))\prime = f\prime (x) \]
\[ g\prime (x) h(x) + g(x) h\prime (x) = f\prime (x) \]
\[ g(x) h\prime (x) = f\prime (x) - g\prime (x) h(x) \]
\[ g(x) h\prime (x) = f\prime (x) - g\prime (x) \frac{f(x)}{g(x)} \]
\[ g(x) h\prime (x) = f\prime (x) - \frac{f(x) g\prime(x)}{g(x)} \]
\[ g(x) h\prime (x) = \frac{f\prime (x) g(x)}{g(x)} - \frac{f(x) g\prime (x)}{g(x)} \]
\[ g(x) h\prime (x) = \frac{f\prime (x) g(x) - f(x) g\prime (x)}{g(x)} \]
\[ h\prime (x) = \frac{f\prime (x) g(x) - f(x) g\prime (x)}{g^2(x)} \]
\[ (\frac{f}{g})\prime = \frac{f\prime g - f g\prime}{g^2} \]
L'Hopital's rule¶
\[ f(c) = 0 \]
\[ g(c) = 0 \]
\[ ? = \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{f(c + dx)}{g(c + dx)} = \frac{f(c) + df_c}{g(c) + dg_c} = \frac{df_c}{dg_c} = \frac{\frac{df_c}{dc}}{\frac{dg_c}{dc}} \]
\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{\frac{d}{dx} f(x)}{\frac{d}{dx} g(x)} \]