2024/G8/3

Section 1
Section 2

3.1 THE LIMIT OF A FUNCTION

$\mathbf {New\ rules}$

${\frac {d}{dx}}[c]=0$

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

${\frac {d}{dx}}[f(x)\pm g(x)]={\frac {d}{dx}}[f(x)]\pm {\frac {d}{dx}}[g(x)]$

${\frac {d}{dx}}[a^{x}]=\ln(a)a^{x}$

${\frac {d}{dx}}[e^{x}]=e^{x}$

$\color {green}Power\,Rule$

${\frac {d}{dx}}[x^{n}]=n\cdot x^{n}-1$

3.2 CALCULATING LIMITS USING THE LIMIT LAWS

$\mathbf {New\ rules}$

$\color {Blue}Product\,Rule$

${\frac {d}{dx}}[f\cdot {g}]={\frac {d}{dx}}[f]\cdot {g}+{\frac {d}{dx}}[g]\cdot {f}$

$\color {Red}Quotient\,Rule$

${\frac {d}{dx}}[{\frac {f}{g}}]={\frac {{\frac {d}{dx}}[f]\cdot {g}-{\frac {d}{dx}}[g]\cdot {f}}{g^{2}}}$

$\mathbf {Examples}$

$\mathbf {Ex.1}$

If $f(x)=x\cdot {e^{x}}$

$f^{\prime }(x)=1\cdot {e^{x}}+x\cdot {e^{x}}$
$f^{\prime }(x)=e^{x}(1+x)$

$\mathbf {Ex.2}$

If $f(t)={\sqrt {t}}(a+bt)$

$f^{\prime }(t)={\frac {1}{2{\sqrt {t}}}}(a+bt)+t{\sqrt {t}}(b)$

$\mathbf {Ex.3}$

If $f(x)={\sqrt {x}}\cdot {g(x)}$

$g(4)=2$

$g^{\prime }(4)=3$

$f^{\prime }(x)={\frac {1}{2{\sqrt {x}}}}\cdot {g(x)}+{\sqrt {x}}\cdot {g^{\prime }(x)}$
$f^{\prime }(x)={\frac {1}{4}}\cdot {2}+{2}\cdot {3}$
$f^{\prime }(x)=6.5$

$\mathbf {Ex.4}$
If $y={\frac {x^{2}+x-2}{x^{3}+6}}$
$y^{\prime }={\frac {(x^{3}+6)(2x+1)-(x^{2}+x-2)(3x^{2})}{(x^{3}+6)^{2}}}$
$y^{\prime }={\frac {(2x^{4}+x^{3}+12x+6)-(3x^{4}+3x^{3}-6x)}{(x^{3}+6)^{2}}}$
$y^{\prime }={\frac {-x^{4}-2x^{3}+6x+12x+6}{(x^{3}+6)^{2}}}$