math: all the derivatives

docs/mhf4u7.md

@@ -2,6 +2,27 @@
 
 The course code for this page is **MHF4U7**.
 
+## Basic math — move later
+
+### Logarithm rules
+
+The logarithm of a product can be rewritten as the sum of two logarithms.
+$$\log_c(ab)=\log_c(a)+\log_c(b)$$
+
+The logarithm of a quotient can be rewritten as the difference of two logarithms.
+$$\log_c\biggr(\frac{a}{b}\biggr)=\log_c(a)-\log_c(b)$$
+
+The exponentials of a logarithm can be brought down to be coefficients.
+$$\log_c(a^n)=n\log_c(a)$$
+
+Some simple values can be easily found.
+
+$$
+a^{\log_a(x)}=x \\
+\log_a(a)=1 \\
+\log_a(1)=0
+$$
+
 ## 3 - Geometry and trigonometry
 
 To find the result of a primary trig ratio, the related acute angle (RAA) should first be found before referring to the CAST rule to determine quadrants before identifying all correct answers in the domain.
@@ -32,6 +53,13 @@ $$
 \tan 2\theta = \frac{2\tan\theta}{1-\tan^2\theta}
 $$
 
+### Euler's number
+
+Euler's number $e$ is a constant irrational number represented as a special limit in calculus.
+$$e=\lim_{x\to ∞}\biggr(1+\frac{1}{x}\biggr)^x$$
+
+The inverse of $e^x$ is $\log_e(x)$, which is known as the **natural logarithm** and can be rewritten as $\ln(x)$ ("lawn x").
+
 ## 4 - Statistics and probability
 
 !!! note "Definition"
@@ -552,6 +580,31 @@ The **chain rule** applies to trigonometric functions and will be applied recurs
 
 Trigonometric identities are not polynomial so values on an interval need to be determined by substituting values between vertical asymptotes and critical points.
 
+### Extended derivative rules
+
+For an **exponential function** where $f(x)=b^x,b≠0$ or $f(x)=b^{g(x)}$, respectively:
+
+$$
+f´(x)=b^x\cdot\ln(b) \\
+f´(g(x))=b^{g(x)}\cdot\ln(b)\cdot g´(x)
+$$
+
+For a **logarithmic function** where $f(x)=\log_b(x)$ or $f(x)=\log_b(g(x))$, respectively:
+
+$$
+f´(x)=\frac{1}{\ln(b)\cdot x} \\
+f´(x)=\frac{g´(x)}{\ln(b)\cdot g(x)}
+$$
+
+From the above base derivatives the derivatives for functions involving $e$ and the **natural logarithm** can be found:
+
+$$
+\frac{d}{dx}e^x=e^x \\
+\frac{d}{dx}e^{g(x)}=e^{g(x)}\cdot g´(x) \\
+\frac{d}{dx}\ln(x)=\frac{1}{x} \\
+\frac{d}{dx}\ln(g(x))=\frac{g´(x)}{g(x)}
+$$
+
 ### Higher order derivatives
 
 The **second derivative** of $f(x)$ is the derivative of the first derivative of $f(x)$, that is, $f´´(x)$.