2024/G1/2

2.2 THE LIMIT OF A FUNCTION

Notes go here for 2.2... example:
$\lim _{z\to z}[f(x)g(x])=c\cdot [\lim _{x\to a}f(x)]$

2.3 CALCULATING LIMITS USING THE LIMIT LAWS

$\lim _{x\to a}=c\cdot [\lim _{x\to a}f(x)]$

2.7 DERIVATIVES AND RATES OF CHANGE

To find the Tangent Line we use $\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}$
We later apply the points on which we want to find the slope. $f(x)=x^{2}f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}=\lim _{h\to 0}{\frac {(x+h)^{2}-x^{2}}{h}}=\lim _{h\to 0}{\frac {x^{2}+2xh+h^{2}-x^{2})}{h}}=\lim _{h\to 0}{\frac {h^{2}+2xh}{h}}=\lim _{h\to 0}{\frac {({\cancel {h}})(h+2x)}{\cancel {h}}}$
$=\lim _{h\to 0}h+2x=(0)+2x$

2.8 THE DERIVATIVE AS A FUNCTION

${\frac {d}{dx}}[c]=0$
${\frac {d}{dx}}[c\cdot f(x)]=c\cdot {\frac {d}{dx}}[f(x)]$

${\frac {d}{dx}}[f(x)\pm g(x)]={\frac {d}{dx}}[f(x)]\pm {\frac {d}{dx}}[g(x)]$
${\frac {d}{dx}}[x^{n}]=n\cdot x^{(n-1)}$
${\frac {d}{dx}}[a^{x}]=\ln(a)a^{x}$
${\frac {d}{dx}}[e^{x}]=e^{x}$