{"id":4258,"date":"2013-12-10T12:05:06","date_gmt":"2013-12-10T03:05:06","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=4258"},"modified":"2014-08-01T19:26:06","modified_gmt":"2014-08-01T10:26:06","slug":"derivatives","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4258","title":{"rendered":"Derivatives"},"content":{"rendered":"
\n

Derivatives<\/h3>\n

The derivative of y = f(x) at a point x is defined as <\/p>\n

\\displaystyle f'(x) = \\lim\\limits_{h \\rightarrow 0}\\frac{f(x+h) - f(x)}{h} = \\lim\\limits_{\\Delta x \\rightarrow 0}\\frac{\\Delta y}{\\Delta x} = \\frac{dy}{dx}<\/p>\n

where h = Δx, Δy = f(x + h) – f(x) = f(x + Δx) – f(x) provided the limit exists. <\/p>\n

Differentiation formulas<\/h3>\n

In the following u, v represent function of x while a, c, p represent constants. It’s assumed that the derivatives of u and v exist, i.e. u and v are differentiable. <\/p>\n

\\displaystyle \\frac{d}{dx}(u \\pm v) = \\frac{du}{dx} \\pm \\frac{dv}{dx}\\\\\\vspace{0.2 in} \\frac{d}{dx}(cu) = c\\frac{du}{dx}\\\\\\vspace{0.2 in} \\frac{d}{dx}(uv) = u\\frac{dv}{dx} + v\\frac{du}{dx}\\\\\\vspace{0.2 in} \\frac{d}{dx}\\left(\\frac{u}{v}\\right) = \\frac{v(du\/dx) - u(dv\/dx)}{v^2}\\\\\\vspace{0.2 in} \\frac{d}{dx}u^p = pu^{p-1}\\frac{du}{dx}\\\\\\vspace{0.2 in} \\frac{d}{dx}(a^u) = a^u\\ln{a}\\\\\\vspace{0.2 in} \\frac{d}{dx}e^u = e^u\\frac{du}{dx}\\\\\\vspace{0.2 in} \\frac{d}{dx}\\ln{u} = \\frac{1}{u}\\frac{du}{dx}\\\\\\vspace{0.2 in} \\frac{d}{dx}\\sin{u} = \\cos{u}\\frac{du}{dx}\\\\\\vspace{0.2 in} \\frac{d}{dx}\\cos{u} = -\\sin{u}\\frac{du}{dx}\\\\\\vspace{0.2 in} \\frac{d}{dx}\\tan{u} = \\sec^2{u}\\frac{du}{dx}\\\\\\vspace{0.2 in} \\frac{d}{dx}\\sin^{-1}u = \\frac{1}{\\sqrt{1 - u^2}}\\frac{du}{dx}\\\\\\vspace{0.2 in} \\frac{d}{dx}\\cos^{-1}u = \\frac{-1}{\\sqrt{1 - u^2}}\\frac{du}{dx}\\\\\\vspace{0.2 in} \\frac{d}{dx}\\tan^{-1}u = \\frac{1}{\\sqrt{1 + u^2}}\\frac{du}{dx}<\/p>\n

In the special case where u = x, the above formulas are simplified since in such case du\/dx = 1. <\/p>\n<\/blockquote>\n