133 lines
2.4 KiB
Markdown
133 lines
2.4 KiB
Markdown
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# Derivation
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The derivative of a function gives us the slope of that function at a specific point.
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# Constant Derivatives
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Deriving any constant gives us a derivative of $x = 0$
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$f(x) = 5$
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$f'(x) = 0$
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# Power Rule
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$\frac{d}{dx}(x^n) = nx^{(n-1)}$
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$f(x) = x^2$
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$f'(x) = x$
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$f(x) = x^5$
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$f'(x) = 5x^4$
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You can also use the power rule to solve $f(x) = \frac{1}{x}$
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$$
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\begin{flalign}
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&f(x) = \frac{1}{x}&\\\
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&f(x) = x^{-1}\\
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&f('x) = -1x^{-1-1} = -1x^{-2} = \frac{-1x^{-2}}{1}\\
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&f'(x) = \frac{-1}{x^2}\\
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\end{flalign}
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$$
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# Constant Multiple Rule
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The derivative of $constant * f(x)$ is $constant * f'(x)$. So we can see that the constant doesn't change.
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$f(x) = 5x^4$
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$f'(x) = 5*4x^3 = 20x^3$
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$f(x) = 8x^4$
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$f'(x) = 32x^3$
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$f(x) = 5x^6$
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$f'(x) = 30x^5$
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# Derive Radical Functions 🤘
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$$
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\begin{flalign}
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&f(x) = \sqrt{x}&\\\
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&f(x) = \sqrt[2]{x^1}\\
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&f(x) = x^{\frac{1}{2}}\\
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&f'(x) = \frac{1}{2}x^{\frac{1}{2}-1} = \frac{1}{2}x^{-\frac{1}{²}}\\
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&f'(x) = \frac{1x^{-\frac{1}{2}}}{2}\\
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&f'(x) = \frac{1}{2x^{\frac{1}{2}}}\\
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&f'(x) = \frac{1}{2\sqrt{x}}
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\end{flalign}
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$$
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# Derive Sine/Cosine
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$\frac{d}{dx}[\sin{x}] = \cos{x}$
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$\frac{d}{dx}[\cos{x}] = -\sin{x}$
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# Product Rule
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This rule applies when you try to derive functions that are multiplied.
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$\frac{d}{dx}[f*x] = f'*g + f*g'$
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# Example
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$$
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\begin{flalign}
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&f(x) = x^3+7x^2-8x+6&\\\
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&f'(x) = 3x^2+14x-8
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\end{flalign}
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$$
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$$
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\begin{flalign}
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&f(x) = 4x^5+3x^4+9x+7&\\\
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&f'(x) = 20x^4+12x^3+9
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\end{flalign}
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$$
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$$
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\begin{flalign}
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&f(x) = 2x^5+5x^3+3x^2+4&\\\
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&\text{Find the slope at } x = 2\\
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&f'(x) = 10x^4+15x^2+6x\\
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&f'(2) = 10(2)^4+15(2)^2+6(2)\\
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&f'(2) = 232\\
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\end{flalign}
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$$
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$$
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\begin{flalign}
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&f(x) = \frac{1}{x^2}&\\\
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&f(x) = x^{-2}\\
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&f'(x) = -2x^{-3}\\
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&f'(x) = \frac{-2}{x^3n }
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\end{flalign}
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$$
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$$
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\begin{flalign}
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&f(x) = \sqrt[3]{x^5}&\\\
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&f(x) = x^{\frac{5}{3}}\\
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&f'(x) = \frac{5}{3}x^{\frac{2}{3}}\\
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&f'(x) = \frac{5x^{\frac{2}{3}}}{3}\\
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&f'(x) = \frac{5\sqrt[3]{x^2}}{3}\\
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\end{flalign}
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$$
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$$
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\begin{flalign}
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&f(x) = \sqrt[7]{x^4}&\\\
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&f(x) = x^{\frac{4}{7}}\\
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&f'(x) = \frac{4}{7}x^{-\frac{3}{7}}\\
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&f'(x) = \frac{4x^{-\frac{3}{7}}}{7}\\
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&f'(x) = \frac{4}{7x^{\frac{3}{7}}}\\
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&f'(x) = \frac{4}{7\sqrt[7]{x^3}}
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\end{flalign}
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$$
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$$
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\begin{flalign}
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&f(x) = (2x-3)^2&\\\
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&f(x) = 4x^2-12x+9\\
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&f'(x) = 8x - 12
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\end{flalign}
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$$
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$$
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\begin{flalign}
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&f(x) = \frac{x^5+6x^4+5x^3}{x^2}&\\\
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&f(x) = x^{-2}(x^5+6x^4+5x^3)\\
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&f(x) = x^3+6x^2+5x\\
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&f'(x) = 3x^2+12x+5
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\end{flalign}
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$$
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