{"id":112,"date":"2020-11-08T04:32:18","date_gmt":"2020-11-08T04:32:18","guid":{"rendered":"https:\/\/www.perfectly-logical.com\/?p=112"},"modified":"2020-11-08T05:39:18","modified_gmt":"2020-11-08T05:39:18","slug":"calculus-references","status":"publish","type":"post","link":"https:\/\/www.perfectly-logical.com\/index.php\/2020\/11\/08\/calculus-references\/","title":{"rendered":"Calculus References"},"content":{"rendered":"\n\n\n

Calculus<\/h2>\n\n\n\n

Special Functions<\/h3>\n\n\n\n

Exponential and Logarithmic Functions<\/h4>\n\n\n\n
\"log_a (x) = y \Leftrightarrow a^y = x\"<\/td><\/tr>
\"ln(x) = log_e (x))\" where \"ln(e) = 1\"<\/td><\/tr>
\"ln (x) = y \Leftrightarrow e^y = x\"<\/td><\/tr><\/tbody><\/table>
Exponential and Logarithmic Functions<\/figcaption><\/figure>\n\n\n\n
Cancellation Equations<\/h5>\n\n\n\n
\"log_a (a^x)=x\"<\/td>\"a^{log_a (x)}=x\"<\/td><\/tr>
\"ln(e^x)=x\"<\/td>\"e^{ln(x)}=x\"<\/td><\/tr><\/tbody><\/table>
Cancellation Equations<\/figcaption><\/figure>\n\n\n\n
Laws of Logarithms<\/h5>\n\n\n\n
\"log_a (xy)=log_a (x) + log_a (y)\"<\/td><\/tr>
\"log_a (\frac{x}{y})=log_a (x) - log_a (y)\"<\/td><\/tr>
\"log_a (x^r)=rlog_a (x)\"<\/td><\/tr><\/tbody><\/table>
Laws of Logarithms<\/figcaption><\/figure>\n\n\n\n

Hyperbolic Functions<\/h4>\n\n\n\n
\"sinh(x) = \frac{e^x - e^{-x}}{2}\"<\/td>\"csch(x) = \frac{1}{sinh(x)}\"<\/td><\/tr>
\"cosh(x) = \frac{e^x + e^{-x}}{2}\"<\/td>\"sech(x) = \frac{1}{cosh(x)}\"<\/td><\/tr>
\"tanh(x) = \frac{sinh(x)}{cosh(x)}\"<\/td>\"coth(x) = \frac{cosh(x)}{sinh(x)}\"<\/td><\/tr><\/tbody><\/table>
Hyperbolic Functions<\/figcaption><\/figure>\n\n\n\n

Inverse Hyperbolic Functions<\/h4>\n\n\n\n
\"y = sinh^{-1} (x) \Leftrightarrow sinh(y) = x\"<\/td>\"sinh^{-1}(x) = ln(x + \sqrt{x^2 + 1})\"<\/td><\/tr>
\"y = cosh^{-1} (x) \Leftrightarrow cosh(y) = x)\" and \"y \geq 0\"<\/td>\"cosh^{-1}(x) = ln(x + \sqrt{x^2 - 1})\"<\/td><\/tr>
\"y = tanh^{-1} (x) \Leftrightarrow tanh(y) = x\"<\/td>\"tanh^{-1}(x) = \frac{1}{2} ln(\frac{1 + x}{1-x})\"<\/td><\/tr><\/tbody><\/table>
Inverse Hyperbolic Functions<\/figcaption><\/figure>\n\n\n\n

Differentiation Rules<\/h3>\n\n\n\n

General Formulas<\/h4>\n\n\n\n
\"\frac{d}{dx} (c) = 0\" where c is a constant<\/td>\"\frac{d}{dx} (c f(x)) = cf^{\prime} (x)\" where c is a constant<\/td><\/tr>
\"\frac{d}{dx} (f(x) + g(x)) = f^{\prime} (x) + g^{\prime} (x)\"<\/td>\"\frac{d}{dx} (f(x) - g(x)) = f^{\prime} (x) - g^{\prime} (x)\"<\/td><\/tr>
\"\frac{d}{dx} (f(x) g(x)) = f^{\prime} (x) g(x) + g^{\prime} (x) f(x)\"
(Product Rule)<\/td>		\"\frac{d}{dx} (\frac{f(x)}{g(x)}) = \frac{f^{\prime} (x) g(x) - g^{\prime} (x) f(x)}{(g(x))^2}\"
(Quotient Rule)<\/td><\/tr>
\"\frac{d}{dx} f(g(x)) = f^{\prime} (g(x)) g^{\prime} (x)\"
(Chain Rule)<\/td>		\"\frac{d}{dx} x^n = nx^{n-1}\"
(Power Rule)<\/td><\/tr><\/tbody><\/table>
General Formulas<\/figcaption><\/figure>\n\n\n\n

Exponential and Logarithmic Functions<\/h4>\n\n\n\n
\"\frac{d}{dx} e^x = e^x\"<\/td>\"\frac{d}{dx} a^x = a^x ln(a)\"<\/td><\/tr>
\"\frac{d}{dx} ln|x| = \frac{1}{x}\"<\/td>\"\frac{d}{dx} log_a (x) = \frac{1}{x ln(a)}\"<\/td><\/tr><\/tbody><\/table>
Exponential and Logarithmic Functions<\/figcaption><\/figure>\n\n\n\n

Trigonometric Functions<\/h4>\n\n\n\n
\"\frac{d}{dx} sin(x) = cos(x)\"<\/td>\"\frac{d}{dx} cos(x) = -sin(x)\"<\/td>\"\frac{d}{dx} tan(x) = sec^2 (x)\"<\/td><\/tr>
\"\frac{d}{dx} csc(x) = -csc(x) cot(x)\"<\/td>\"\frac{d}{dx} sec(x) = sec(x) tan(x)\"<\/td>\"\frac{d}{dx} cot(x) = -csc^2 (x)\"<\/td><\/tr><\/tbody><\/table>
Trigonometric Functions<\/figcaption><\/figure>\n\n\n\n

Inverse Trigonometric Functions<\/h4>\n\n\n\n
\"\frac{d}{dx} sin^{-1} (x) = \frac{1}{\sqrt{1 -x^2}}\"<\/td>\"\frac{d}{dx} cos^{-1} (x) = - \frac{1}{\sqrt{1 -x^2}}\"<\/td>\"\frac{d}{dx} tan^{-1} (x) = \frac{1}{1 + x^2}\"<\/td><\/tr>
\"\frac{d}{dx} csc^{-1} (x) = - \frac{1}{x \sqrt{x^2 - 1}}\"<\/td>\"\frac{d}{dx} sec^{-1} (x) = \frac{1}{x \sqrt{x^2 - 1}}\"<\/td>\"\frac{d}{dx} cot^{-1} (x) = - \frac{1}{1 + x^2}\"<\/td><\/tr><\/tbody><\/table>
Inverse Trigonometric Functions<\/figcaption><\/figure>\n\n\n\n

Hyperbolic Functions<\/h4>\n\n\n\n
\"\frac{d}{dx} sinh(x) = cosh(x)\"<\/td>\"\frac{d}{dx} cosh(x) = sinh(x)\"<\/td>\"\frac{d}{dx} tanh(x) = sech^2 (x)\"<\/td><\/tr>
\"\frac{d}{dx} csch(x) = -csch(x) coth(x)\"<\/td>\"\frac{d}{dx} sech(x) = -sech(x) tanh(x)\"<\/td>\"\frac{d}{dx} coth(x) = -csch^2 (x)\"<\/td><\/tr><\/tbody><\/table>
Hyperbolic Functions<\/figcaption><\/figure>\n\n\n\n

Inverse Hyperbolic Functions<\/h4>\n\n\n\n
\"\frac{d}{dx} sinh^{-1} (x) = \frac{1}{\sqrt{1 + x^2}}\"<\/td>\"\frac{d}{dx} cosh^{-1} (x) = \frac{1}{\sqrt{x^2 - 1}}\"<\/td>\"\frac{d}{dx} tanh^{-1} (x) = \frac{1}{1 - x^2}\"<\/td><\/tr>
\"\frac{d}{dx} csch^{-1} (x) = - \frac{1}{|x| \sqrt{x^2 + 1}}\"<\/td>\"\frac{d}{dx} sech^{-1} (x) = - \frac{1}{x \sqrt{1 - x^2}}\"<\/td>\"\frac{d}{dx} coth^{-1} (x) = \frac{1}{1 - x^2}\"<\/td><\/tr><\/tbody><\/table>
Inverse Hyperbolic Functions<\/figcaption><\/figure>\n\n\n\n

Integrals<\/h3>\n\n\n\n

Let c be a constant<\/p>\n\n\n\n

\"\int u dv = uv - \int v dv\"<\/td>\"\int csc(u) cot(u) du = -csc(u) + c\"<\/td><\/tr>
\"\int u^n du = \frac{u^{n+1}}{n+1} + c\" where \"n\neq-1\"<\/td>\"\int tan(u) du = ln|sec(u)| + c\"<\/td><\/tr>
\"\int \frac{du}{u} = ln|u| + c\"<\/td>\"\int cot(u) du = ln|sin(u)| + c\"<\/td><\/tr>
\"\int e^u du = e^u +c\"<\/td>\"\int sec(u) du = ln|sec(u) + tan(u)| + c\"<\/td><\/tr>
\"\int a^u du = \frac{a^u}{ln(a)} + c\"<\/td>\"\int csc(u) du = ln|csc(u) - cot(u)| + c\"<\/td><\/tr>
\"\int sin(u) du = -cos(u) + c\"<\/td>\"\int \frac{du}{\sqrt{a^2 - u^2}} du = sin^{-1} (\frac{u}{a}) + c\"<\/td><\/tr>
\"\int cos(u) du = sin(u) + c\"<\/td>\"\int \frac{du}{a^2 + u^2} du = \frac{1}{a} tan^{-1} (\frac{u}{a}) + c\"<\/td><\/tr>
\"\int sec^2 (u) du = tan(u) + c\"<\/td>\"\int \frac{du}{u \sqrt{u^2 - a^2}} du = \frac{1}{a} sec^{-1} (\frac{u}{a}) + c\"<\/td><\/tr>
\"\int csc^2 (u) du = -cot(u) + c\"<\/td>\"\int \frac{du}{a^2 - u^2} du = \frac{1}{2a} ln|\frac{u+a}{u-a}| + c\"<\/td><\/tr>
\"\int sec(u) tan(u) du = sec(u) + c\"<\/td>\"\int \frac{du}{u^2 - a^2} du = \frac{1}{2a} ln|\frac{u-a}{u+a}| + c\"<\/td><\/tr><\/tbody><\/table>
Integrals<\/figcaption><\/figure>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

Some useful formulas to help when doing calculus<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,29],"tags":[15,8,12],"class_list":["post-112","post","type-post","status-publish","format-standard","hentry","category-math","category-reference","tag-formulas","tag-math","tag-reference"],"_links":{"self":[{"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/posts\/112","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/comments?post=112"}],"version-history":[{"count":7,"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/posts\/112\/revisions"}],"predecessor-version":[{"id":119,"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/posts\/112\/revisions\/119"}],"wp:attachment":[{"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/media?parent=112"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/categories?post=112"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/tags?post=112"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}