Calculus
Derivatives¶
\[ \frac{f(x + dx) - f(x)}{dx} = \frac{df}{dx} = f\prime (x) \]
sum rule¶
\[ \frac{d(f(x) + g(x))}{dx} = \frac{f(x + dx) + g(x + dx) - f(x) -g(x)}{dx} = \frac{f(x + dx) - f(x)}{dx} + \frac{g(x + dx) - g(x)}{dx} = \frac{df}{dx} + \frac{dg}{dx} \]
\[ \frac{d(f(x) + g(x))}{dx} = \frac{df}{dx} + \frac{dg}{dx} \]
\[ (f + g)\prime = f\prime + g\prime \]
product rule¶
\[ f(x + dx) = f(x) + df \]
\[ \frac{d(f(x)g(x))}{dx} = \frac{(f(x) + df)(g(x) + dg) - f(x)g(x)}{dx} = \frac{f(x)g(x) + f(x)dg + dfg(x) + dfdg - f(x)g(x)}{dx} = f(x)\frac{dg}{dx} + \frac{df}{dx}g(x) + \frac{dfdg}{dx} \]
\[ \frac{dfdg}{dx} \to 0 \]
\[ (fg)\prime = fg\prime + f\prime g \]
chain rule¶
\[ \frac{f(x + \Delta x) - f(x)}{\Delta x} = \frac{df}{dx} (x) \]
\[ \Delta x \to 0 \]
\[ \frac{d(f(g(x)))}{dx} = \frac{f(g(x + dx)) - f(g(x))}{dx} = \frac{f(g(x) + dg) - f(g(x))}{dx} \]
\[ dg \to 0 \]
\[ \frac{f(g(x) + dg) - f(g(x))}{dg} = \frac{df}{dx}(g(x)) \]
\[ \frac{f(g(x) + dg) - f(g(x))}{dg} \frac{dg}{dx} = \frac{df}{dx}(g(x)) \frac{dg}{dx} = \frac{f(g(x) + dg) - f(g(x))}{dx} = \frac{d(f(g(x)))}{dx} \]
\[ \frac{d(f(g(x)))}{dx} = \frac{df}{dx}(g(x))\frac{dg}{dx} \]
\[ (f(g))\prime = f\prime(g) g\prime \]
mbc rule¶
\[ \frac{dcf(x)}{dx} = c f\prime(x) \]
\[ (cf)\prime = c f\prime \]
exponent rule¶
\[ \frac{d(a^x)}{dx} = \frac{a^{x + dx} -a^x}{dx} = \frac{a^x a^{dx} - a^x}{dx} = a^x \frac{a^{dx} - 1}{dx} \]
\[ \frac{d(a^x)}{dx} = a^x \frac{a^{dx} - 1}{dx} \]
\[ \text{(lets figure this out later!)} \]
e¶
\[ \frac{de^x}{dx} = e^x = e^x \frac{e^{dx} - 1}{dx} \]
\[ \frac{e^{dx} - 1}{dx} = 1 \]
\[ e^{dx} -1 = dx \]
\[ e^{dx} = 1 + dx \]
\[ e = (1 + dx)^{\frac{1}{dx}} \]
\[ log_e (x) = ln(x) \]
logarithmic derivitave¶
\[ e^{ln(f(x))} = f(x) \]
\[ \frac{d(e^{ln(f(x)})}{dx} = (ln(f(x)))\prime e^{ln(f(x))} = f(x) (ln(f(x)))\prime = f\prime(x) \]
\[ (ln(f))\prime = \frac{f\prime}{f} \]
\[ f(x) = a^x \]
\[ ln(f(x)) = ln(a^x) = x ln(a) \]
\[ \frac{f\prime (x)}{f(x)} = ln(a) \]
\[ (a^x)\prime = a^x ln(a) \]
power rule¶
\[ f(x) = x^n \]
\[ ln(f(x)) = ln(x^n) = n ln(x) \]
\[ ln\prime(x) = \frac{1}{x} \]
\[ \frac{f\prime(x)}{f(x)} = \frac{n}{x} \]
\[ (x^n)\prime = x^n \frac{n}{x} \]
\[ (x^n)\prime = n x^{n - 1} \]
\[ \text{in exactly } 100 \text{ lines.} \]