User:JaviPrieto/Derivatives


 * $$m={\mbox{change in } y \over \mbox{change in } x} = {\Delta y \over{\Delta x}}$$
 * $$ \frac{dy}{dx} \,\!$$
 * $$m = \frac{\Delta f(x)}{\Delta x} = \frac{f(x+h)-f(x)}{h}.$$
 * $$f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$$
 * $$\lim_{h\to 0}{f(a+h)-f(a) - f'(a)\cdot h\over h} = 0,$$
 * $$f(a+h) \approx f(a) + f'(a)h$$
 * $$Q(h) = \frac{f(a + h) - f(a)}{h}.$$

Q(h) is the slope of the secant line between (a, ç'(a)) and (a + h, ç'(a + h)). If ç' is a continuous function, meaning that its graph is an unbroken curve with no gaps, then Q is a continuous function away from the point h'' = 0. If the limit $$\textstyle\lim_{h\to 0} Q(h)$$ exists, meaning that there is a way of choosing a value for Q(0) that makes the graph of Q a continuous function, then the function ç' is differentiable at the point a, and its derivative at a equals Q''(0).
 * $$f'(3)= \lim_{h\to 0}{f(3+h)-f(3)\over h} = \lim_{h\to 0}{(3+h)^2 - 9\over{h}} = \lim_{h\to 0}{9 + 6h + h^2 - 9\over{h}} = \lim_{h\to 0}{6h + h^2\over{h}} = \lim_{h\to 0}{6 + h}. $$
 * $$ \lim_{h\to 0}{6 + h} = 6 + 0 = 6. $$
 * $$\begin{align} 1 &{}\mapsto 2,\\ 2 &{}\mapsto 4,\\ 3 &{}\mapsto 6.\end{align}$$
 * $$\begin{align} D(x \mapsto 1) &= (x \mapsto 0),\\ D(x \mapsto x) &= (x \mapsto 1),\\ D(x \mapsto x^2) &= (x \mapsto 2\cdot x).\end{align}$$
 * $$x \mapsto x^2,$$
 * $$ x \mapsto 2x ,$$
 * $$f(x) = \begin{cases} x^2, & \mbox{if }x\ge 0 \\ -x^2, & \mbox{if }x \le 0.\end{cases}$$
 * $$f'(x) = \begin{cases} 2x, & \mbox{if }x\ge 0 \\ -2x, & \mbox{if }x \le 0.\end{cases}$$
 * $$ f(x+h) \approx f(x) + f'(x)h + \tfrac12 f''(x) h^2$$
 * $$ \lim_{h\to 0}\frac{f(x+h) - f(x) - f'(x)h - \frac12 f''(x) h^2}{h^2}=0.$$
 * $$\frac{dy}{dx},\quad\frac{d f}{dx}(x),\;\;\mathrm{or}\;\; \frac{d}{dx}f(x),$$
 * $$\frac{d^ny}{dx^n},\quad\frac{d^n f}{dx^n}(x),\;\;\mathrm{or}\;\;\frac{d^n}{dx^n}f(x)$$
 * $$\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right).$$
 * $$\left.\frac{dy}{dx}\right|_{x=a} = \frac{dy}{dx}(a).$$
 * $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.$$
 * $$(f')'=f\,$$ &emsp; and &emsp; $$(f)'=f'''.\,$$
 * $$f^{\mathrm{iv}}\,\!$$ &emsp; or &emsp; $$f^{(4)}.\,\!$$
 * $$\dot{y}$$ &emsp; and &emsp; $$\ddot{y}$$
 * $$D_x y\,$$ &emsp; or &emsp; $$D_x f(x)\,$$,
 * $$ f(x) = x^r,\,$$
 * $$ f'(x) = rx^{r-1},\,$$

wherever this function is defined. For example, if $$f(x) = x^{1/4}$$, then
 * $$f'(x) = (1/4)x^{-3/4},\,$$
 * $$ \frac{d}{dx}e^x = e^x$$
 * $$ \frac{d}{dx}a^x = \ln(a)a^x$$
 * $$ \frac{d}{dx}\ln(x) = \frac{1}{x},\qquad x > 0$$
 * $$ \frac{d}{dx}\log_a(x) = \frac{1}{x\ln(a)}$$
 * $$ \frac{d}{dx}\sin(x) = \cos(x).$$
 * $$ \frac{d}{dx}\cos(x) = -\sin(x).$$
 * $$ \frac{d}{dx}\tan(x) = \sec^2(x) = \frac{1}{\cos^2(x)} = 1+\tan^2(x).$$
 * $$ \frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1-x^2}}.$$
 * $$ \frac{d}{dx}\arccos(x)= -\frac{1}{\sqrt{1-x^2}}.$$
 * $$ \frac{d}{dx}\arctan(x)= \frac{1}.$$
 * $$f' = 0 \,$$
 * $$(af + bg)' = af' + bg' \,$$ for all functions &fnof; and g and all real numbers a and b.
 * $$(fg)' = f 'g + fg' \,$$ for all functions &fnof; and g.
 * $$\left(\frac{f}{g} \right)' = \frac{f'g - fg'}{g^2}$$ for all functions &fnof; and g where g ? 0.


 * Chain rule: If $$f(x) = h(g(x))$$, then
 * $$f'(x) = h'(g(x)) \cdot g'(x). \,$$
 * $$f(x) = x^4 + \sin (x^2) - \ln(x) e^x + 7\,$$
 * $$\begin{align}f'(x) &= 4 x^{(4-1)}+ \frac{d\left(x^2\right)}{dx}\cos (x^2) - \frac{d\left(\ln {x}\right)}{dx} e^x - \ln{x} \frac{d\left(e^x\right)}{dx} + 0 \\     &= 4x^3 + 2x\cos (x^2) - \frac{1}{x} e^x - \ln(x) e^x.\end{align}$$
 * $$\mathbf{y}'(t) = (y'_1(t), \ldots, y'_n(t)).$$
 * $$\mathbf{y}'(t)=\lim_{h\to 0}\frac{\mathbf{y}(t+h) - \mathbf{y}(t)}{h},$$
 * $$y'_1(t)\mathbf{e}_1 + \cdots + y'_n(t)\mathbf{e}_n$$
 * $$f(x,y) = x^2 + xy + y^2.\,$$
 * $$f(x,y) = f_x(y) = x^2 + xy + y^2.\,$$
 * $$x \mapsto f_x,\,$$
 * $$f_x(y) = x^2 + xy + y^2.\,$$
 * $$f_a(y) = a^2 + ay + y^2.\,$$
 * $$f_a'(y) = a + 2y.\,$$
 * $$\frac{\partial f}{\partial y}(x,y) = x + 2y.$$
 * $$\frac{\partial f}{\partial x_i}(a_1,\ldots,a_n) = \lim_{h \to 0}\frac{f(a_1,\ldots,a_i+h,\ldots,a_n) - f(a_1,\ldots,a_n)}{h}.$$
 * $$f_{a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_n}(x_i) = f(a_1,\ldots,a_{i-1},x_i,a_{i+1},\ldots,a_n)$$
 * $$\frac{df_{a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_n}}{dx_i}(a_i) = \frac{\partial f}{\partial x_i}(a_1,\ldots,a_n).$$
 * $$\nabla f(a) = \left(\frac{\partial f}{\partial x_1}(a), \ldots, \frac{\partial f}{\partial x_n}(a)\right).$$
 * $$\mathbf{v} = (v_1,\ldots,v_n).$$
 * $$D_{\mathbf{v}}{f}(\mathbf{x}) = \lim_{h \rightarrow 0}{\frac{f(\mathbf{x} + h\mathbf{v}) - f(\mathbf{x})}{h}}.$$
 * $$\frac{f(\mathbf{x} + (k/\lambda)(\lambda\mathbf{u})) - f(\mathbf{x})}{k/\lambda}= \lambda\cdot\frac{f(\mathbf{x} + k\mathbf{u}) - f(\mathbf{x})}{k}.$$
 * $$D_{\mathbf{v}}{f}(\boldsymbol{x}) = \sum_{j=1}^n v_j \frac{\partial f}{\partial x_j}.$$
 * $$f(\mathbf{a} + \mathbf{v}) \approx f(\mathbf{a}) + f'(\mathbf{a})\mathbf{v}.$$
 * $$f(\mathbf{a} + \mathbf{v}) - f(\mathbf{a}) \approx f'(\mathbf{a})\mathbf{v}.$$
 * $$f(\mathbf{a} + \mathbf{v} + \mathbf{w}) - f(\mathbf{a} + \mathbf{v}) - f(\mathbf{a} + \mathbf{w}) + f(\mathbf{a})\approx f'(\mathbf{a} + \mathbf{v})\mathbf{w} - f'(\mathbf{a})\mathbf{w}.$$
 * $$\begin{align}0&\approx f(\mathbf{a} + \mathbf{v} + \mathbf{w}) - f(\mathbf{a} + \mathbf{v}) - f(\mathbf{a} + \mathbf{w}) + f(\mathbf{a}) \\&= (f(\mathbf{a} + \mathbf{v} + \mathbf{w}) - f(\mathbf{a})) - (f(\mathbf{a} + \mathbf{v}) - f(\mathbf{a})) - (f(\mathbf{a} + \mathbf{w}) - f(\mathbf{a})) \\&\approx f'(\mathbf{a})(\mathbf{v} + \mathbf{w}) - f'(\mathbf{a})\mathbf{v} - f'(\mathbf{a})\mathbf{w}.\end{align}$$
 * $$\lim_{h \to 0} \frac{f(a + h) - f(a) - f'(a)h}{h} = 0.$$
 * $$\lim_{h \to 0} \frac{|f(a + h) - f(a) - f'(a)h|}{|h|} = 0$$
 * $$\lim_{\lVert\mathbf{h}\rVert \to 0} \frac{\lVert f(\mathbf{a} + \mathbf{h}) - f(\mathbf{a}) - f'(\mathbf{a})\mathbf{h}\rVert}{\lVert\mathbf{h}\rVert} = 0.$$
 * $$f'(\mathbf{a}) = \operatorname{Jac}_{\mathbf{a}} = \left(\frac{\partial f_i}{\partial x_j}\right)_{ij}.$$
 * $$f(a+h) \approx f(a) + f'(a)h.$$

Up to changing variables, this is the statement that the function $$x \mapsto f(a) + f'(a)(x-a)$$ is the best linear approximation to ç' at a''.