Derivatives¶
At time \(t\), the derivative \(f'(t)\) or \(df/dt\) or \(v(t)\) is: \(\(\begin{equation} f'(t) = \lim_{\Delta_{t}\rightarrow{0}}\frac{f(t + \Delta_{t}) - f(t)}{\Delta{t}} \end{equation}\)\)
The derivative is a function, \(f'(x)\).
Power Rule¶
\(f(x) = x^{n}, n \neq 0\) \(\(f'(x) = n \cdot x^{n-1}\)\)
Examples:¶
\(f(x) = x^{2}\) \(f'(x) = 2x\)
\(h(x) = x^{-100}\) \(h'(x) = -100x^{-100-1} = -100x^{-101}\)
\(z(x) = x^{2.571}\) \(z'(x) = 2.571x^{1.571}\)
Basic Derivative Rules¶
\(\frac{\partial}{\partial{x}} A = 0\) where \(A\) is a constant.
\(\frac{\partial}{\partial{x}} A \cdot f(x) = A \frac{\partial}{\partial{x}} f(x) = A f'(x)\)
- \(\frac{\partial}{\partial{x}} [2x^{5}] = 2 \frac{\partial}{\partial{x}} [x^{5}] = 2 \cdot 5x^{4} = 10x^{4}\)
\(\frac{\partial}{\partial{x}} [f(x) + g(x)] = f'(x) + g'(x)\)
- \(\frac{\partial}{\partial{x}} [ x^{3} + x^{-4}] = 3x^{2} - 4x^{-5}\)
Any polynomial¶
\(f(x) = 2x^{3} - 7x^{2} + 3x - 100\)
\(f'(x) = 2 \cdot 3x^{2} - 7 \cdot 2x + 3\) * \(f'(x) = 6x^{2} - 14x + 3\)
\(f(x) = x^{5} + 2x^{3} - x^{2}\)
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\(\frac{\partial}{\partial{x}}f(x) = \frac{\partial}{\partial{x}}[ x^{5} + 2x^{3} - x^{2} ]\)
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\(f'(x) = \frac{\partial}{\partial{x}}(x^{5}) + \frac{\partial}{\partial{x}}(2x^{3}) - \frac{\partial}{\partial{x}}(x^{2})\)
\(f'(x) = 5x^{4} + 6x^{2} - 2x\)
- \(f'(2) = 5 \cdot 2^{4} + 6 \cdot 2^{2} - 4\)
- \(f'(2) = 80 + 24 - 4 = 100\)
Sine & Cosine Derivatives¶
Euler's Number¶
- The slope of the tangent line to \(y=e^{x}\) is \(e^{x} = f'(x)\).
\(\ln (x)\)¶
Product Rule¶
\(\frac{\partial}{\partial{x}}[x^{2} \sin{x}] = 2x \cdot \sin{x} + x^{2} \cdot \cos{x}\)
Chain Rule¶
\(h(x) = (\sin{x})^{2}\)
\(\frac{\partial}{\partial{(\sin{x})}} [(\sin{x})^2] = 2(\sin{x})\)
\(\frac{\partial}{\partial{x}} [\sin{x}] = \cos{x}\)
Derivative of inner function times the derivative of outer function.