| Front | Theorem (14.2.1): Properties of the Derivative of Vector Functions |
| Back | If $\overrightarrow{f}$, $\overrightarrow{g}$, and $u$ are continuous on a common domain, then$$(1) \overrightarrow{(f+g)'(t)} = \overrightarrow{f'(t)} + \overrightarrow{g'(t)}$$$$(2) (\overrightarrow{(\alpha f)'(t)} = \alpha \overrightarrow{f'(t)} \text{ and } \overrightarrow(uc)'(t) = u'(t)\overrightarrow{c}$$$$(3) \overrightarrow{(uf)'(t)} = u(t)\overrightarrow{f'(t)} + u'(t)\overrightarrow{f(t)}$$$$(4) (\overrightarrow{f} \cdot \overrightarrow{g})'(t) = [\overrightarrow{f(t)} \cdot \overrightarrow{g'(t)}] + [\overrightarrow{f'(t)} \cdot \overrightarrow{g(t)]$$$$(5) (\overrightarrow{f} \times \overrightarrow{g})'(t) = [\overrightarrow{f(t)} \times \overrightarrow{g'(t)}] + [\overrightarrow{f'(t)} \times \overrightarrow{g(t)}]$$$$(6) \overrightarrow{(f \circ u)'(t)} = \overrightarrow{f'(u(t))}u'(t) = u'(t) \overrightarrow{f'(u(t))}$$ |
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| Front | Theorem (14.3.4): Unit Tangent Vector |
| Back | $$\overrightarrow{T(t)} = \frac{\overrightarrow{r'(t)}}{||\overrightarrow{r'(t)}||}$$ |
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| Front | Definition (15.4.1): Partial Derivatives (Two Variables) |
| Back | Let $f$ be a function of two variables x, y. The partial derivatives of $f$ with respect to $x$ and with respect to $y$ are the functions $$f_{x}(x, y) = \lim_{h\to0}\frac{f(x+h, y) - f(x,y)}{h}$$ $$f_{y}(x, y) = \lim_{h\to0}\frac{f(x, y+h) - f(x,y)}{h}$$ provided that these limits exist. |
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