2024/G2/12

$\mathbf {3.1}$
1. The derivative of a constant is 0 = ${\frac {d}{dx}}[c]=0$
2. ${\frac {d}{dx}}[x^{n}]=nx^{n-1}$

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

4. ${\frac {d}{dx}}[f(x)+g(x)]={\frac {d}{dx}}[f(x)]+{\frac {d}{dx}}[g(x)]$

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

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


$\mathbf {\color {Blue}{Examples}}$

1) ${\frac {d}{dx}}[5+\pi ]=0+0=0$

2) ${\frac {d}{dx}}[3x]=3\cdot {\frac {d}{dx}}[x]=3(1)(x)^{1-1}=3\cdot x^{0}=3$

3) ${\frac {d}{dx}}[9x^{2}]=18x$

4) ${\frac {d}{dx}}[x^{3}+x^{2}+10]=3x^{2}+2x+0$

5) ${\frac {d}{dx}}[{\sqrt {(}}x)]={\frac {d}{dx}}[x^{\frac {1}{2}}]={\frac {1}{2x}}^{{\frac {1}{2}}-{\frac {2}{2}}}={\frac {1}{2x}}^{\frac {-1}{2}}={\frac {1}{2{\sqrt {(}}x)}}$

A1. ${{\sqrt[{n}]{(}}x^{m})}=({\sqrt[{n}]{x}})^{m}=x^{\frac {m}{n}}$

6) ${\frac {d}{dx}}[{{\sqrt[{3}]{(}}x^{2})}]={\frac {d}{dx}}[x^{\frac {2}{3}}]={\frac {2}{3}}\cdot x^{\frac {-1}{3}}={{\frac {2}{3}}\cdot {\sqrt[{3}]{(}}x)}$

7) ${\frac {d}{dx}}[({\sqrt[{5}]{x}})^{7}]={\frac {d}{dx}}[{\frac {7}{x^{5}}}]={\frac {7}{5}}\cdot x^{{\frac {7}{5}}-1}={\frac {7}{5}}\cdot x^{\frac {2}{5}}={\frac {7}{5}}\cdot {{\sqrt[{5}]{x}}^{2}}$

8) ${\frac {d}{dx}}[3x^{10}+e^{x}-5^{x}]=30x^{9}+e^{x}-ln(5)\cdot 5^{x}$

9) ${\frac {d}{dx}}[{\frac {1}{x}}]={\frac {d}{dx}}[x^{-1}]=-1\cdot x^{-1-1}=-x^{-2}=-{\frac {1}{x^{2}}}$

$\mathbf {\color {Purple}{PracticeProblems}}$

1) ${\frac {d}{dx}}[\pi +3x-5x^{3}]=0+3-15x^{2}$

2) ${\frac {d}{dx}}[e^{x}+3^{x}-5\cdot 2^{x}]=e^{x}+ln(3)\cdot 3^{x}-5\cdot ln(2)\cdot 2^{x}$

3) ${\frac {d}{dx}}[x^{5}-6x^{3}+2x+5]=5x^{4}-18x^{2}+2+0$

4) ${\frac {d}{dx}}[x{\sqrt {x}}]={\frac {d}{dx}}[x^{1}\cdot x^{\frac {1}{2}}]={\frac {d}{dx}}[x^{\frac {3}{2}}]={\frac {3}{2}}\cdot x^{{\frac {3}{2}}-1}={\frac {3}{2x}}^{\frac {3}{2}}-{\frac {2}{2}}={\frac {3}{2x}}^{\frac {1}{2}}={\frac {3}{2}}\cdot {\sqrt {x}}$

5) Failed to parse (syntax error): {\displaystyle {\frac{d}{dx}} [{\frac{3}{x}} + \sqrt 2 - 5^{x}] = 3 \cdot {\frac{d}{dx}} [{\frac{1}{x}}] = 3 \cdot {\frac{d}{dx}} [x^{-1}] = (-1)(3) \cdot x^{-1-1} = -3 \cdot x^{-2} = {\frac{3}{2x}} + 0 -ln(-5) \cdot -5^{x} = -{\frac{3}{x^{2}} - ln(-5) \cdot -5^{x} }