2024/G2/12: Difference between revisions

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1) <math>{\frac{d}{dx}} [5 + \pi] = 0+0=0 </math> <br><br>
1) <math>{\frac{d}{dx}} [5 + \pi] = 0+0=0 </math> <br><br>
2) <math>{\frac{d}{dx}} [3x] = 3 \cdot {\frac{d}{dx}} [x] = 3(1)(x)^{1-1} = 3 \cdot x^0 = 3 </math> <br><br>
2) <math>{\frac{d}{dx}} [3x] = 3 \cdot {\frac{d}{dx}} [x] = 3(1)(x)^{1-1} = 3 \cdot x^0 = 3 </math> <br><br>
3) <math>{\frac{d}{dx}} [9x^2] = 18x </math> <br><br>
3) <math>{\frac{d}{dx}} [9x^2] = 18x </math> <br><br>
4) <math>{\frac{d}{dx}} [x^3 + x^2 + 10] = 3x^2 +2x + 0 </math> <br>
 
5) <math>{\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)}}</math><br>
4) <math>{\frac{d}{dx}} [x^3 + x^2 + 10] = 3x^2 +2x + 0 </math> <br><br>
A1. <math> {\sqrt[n](x^m)} = ({\sqrt[n]x})^m = x^{\frac{m}{n}} </math><br>
 
6) <math> {\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)} </math><br>
5) <math>{\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)}}</math><br><br>
7) <math> {\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} </math><br>
 
8) <math> {\frac{d}{dx}} [3x^{10}+e^{x}-5^{x}] = 30x^{9} + e^{x} - ln(5) \cdot 5^{x} </math><br>
A1. <math> {\sqrt[n](x^m)} = ({\sqrt[n]x})^m = x^{\frac{m}{n}} </math><br><br>
9) <math> {\frac{d}{dx}} [{\frac{1}{x}}] = {\frac{d}{dx}} [x^{-1}] = -1 \cdot x^{-1-1} = -x^{-2} = - {\frac{1}{x^2}} </math><br>
 
6) <math> {\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)} </math><br><br>
 
7) <math> {\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} </math><br><br>
 
8) <math> {\frac{d}{dx}} [3x^{10}+e^{x}-5^{x}] = 30x^{9} + e^{x} - ln(5) \cdot 5^{x} </math><br><br>
 
9) <math> {\frac{d}{dx}} [{\frac{1}{x}}] = {\frac{d}{dx}} [x^{-1}] = -1 \cdot x^{-1-1} = -x^{-2} = - {\frac{1}{x^2}} </math><br><br><br>
 
<math>\mathbf{\color{Purple}{PracticeProblems}}</math><br>
 
1) <math> {\frac{d}{dx}} [\pi + 3x -5x^{3}] = 0+3-15x^{2} </math><br><br>
 
2) <math> {\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} </math><br><br>
 
3) <math> {\frac{d}{dx}} [x^{5} - 6x^{3} +2x +5] = 5x^{4} -18x^{2} + 2 + 0 </math><br><br>
 
4) <math> {\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 </math><br><br>
 
5) <math> {\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}{x^{2}}} + 0 - ln(-5) \cdot -5^{x} = -{\frac{3}{x^{2}}} - ln(-5) \cdot -5^{x}  </math> <br><br>
 
6) <math> {\frac{d}{dx}} [{\frac{4}{\sqrt[3]x^{2}}} - 3x^{9} + \sqrt[3]x \cdot {\frac{1}{x}}] = 4 \cdot x^{-{\frac{2}{3}}} = x^{-{\frac{2}{3}}-{\frac{3}{3}}} = x^{-{\frac{5}{3}}} </math><br><br>

Latest revision as of 16:45, 14 April 2023

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{3.1}}
1. The derivative of a constant is 0 = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [c] = 0 }
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [x^n] = nx^{n-1} }

3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [c\cdot f(x)] = c \cdot {\frac{d} {dx}} [f(x)] }

4. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [f(x) + g(x)] = {\frac{d}{dx}} [f(x)] + {\frac{d}{dx}} [g(x)] }

5. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [a^x] = ln (a) \cdot a^x }

6. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [e^x] = e^x }


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{\color{Blue}{Examples}}}

1) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [5 + \pi] = 0+0=0 }

2) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [3x] = 3 \cdot {\frac{d}{dx}} [x] = 3(1)(x)^{1-1} = 3 \cdot x^0 = 3 }

3) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [9x^2] = 18x }

4) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [x^3 + x^2 + 10] = 3x^2 +2x + 0 }

5) Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\sqrt[n](x^m)} = ({\sqrt[n]x})^m = x^{\frac{m}{n}} }

6) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\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) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\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) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [3x^{10}+e^{x}-5^{x}] = 30x^{9} + e^{x} - ln(5) \cdot 5^{x} }

9) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [{\frac{1}{x}}] = {\frac{d}{dx}} [x^{-1}] = -1 \cdot x^{-1-1} = -x^{-2} = - {\frac{1}{x^2}} }


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{\color{Purple}{PracticeProblems}}}

1) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [\pi + 3x -5x^{3}] = 0+3-15x^{2} }

2) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\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) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [x^{5} - 6x^{3} +2x +5] = 5x^{4} -18x^{2} + 2 + 0 }

4) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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}{x^{2}}} + 0 - ln(-5) \cdot -5^{x} = -{\frac{3}{x^{2}}} - ln(-5) \cdot -5^{x} }

6) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}} [{\frac{4}{\sqrt[3]x^{2}}} - 3x^{9} + \sqrt[3]x \cdot {\frac{1}{x}}] = 4 \cdot x^{-{\frac{2}{3}}} = x^{-{\frac{2}{3}}-{\frac{3}{3}}} = x^{-{\frac{5}{3}}} }

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