{"id":8648,"date":"2024-01-06T20:05:26","date_gmt":"2024-01-06T14:35:26","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=8648"},"modified":"2024-01-06T20:05:37","modified_gmt":"2024-01-06T14:35:37","slug":"derivatives-of-hyperbolic-functions","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/derivatives-of-hyperbolic-functions\/","title":{"rendered":"Derivatives of hyperbolic functions"},"content":{"rendered":"\n

The derivatives of the hyperbolic functions are quite straightforward and somewhat analogous to the derivatives of their trigonometric counterparts. Here are the derivatives for the six primary hyperbolic functions:<\/p>\n\n\n\n

(1) Hyperbolic Sine $sinh$:<\/strong>
$$
\\frac{d}{dx} \\sinh(x) = \\cosh(x)
$$
The derivative of hyperbolic sine is hyperbolic cosine.<\/p>\n\n\n\n

Proof<\/strong>
The hyperbolic sine and cosine functions are defined as follows:<\/p>\n\n\n\n

$$
\\sinh(x) = \\frac{e^x – e^{-x}}{2}
$$
$$
\\cosh(x) = \\frac{e^x + e^{-x}}{2}
$$<\/p>\n\n\n\n

Now,<\/p>\n\n\n\n

$$
\\frac{d}{dx} \\sinh(x) = \\frac{d}{dx} \\left( \\frac{e^x – e^{-x}}{2} \\right)
$$<\/p>\n\n\n\n

$$
= \\frac{1}{2}\\left( \\frac{d}{dx}(e^x) – \\frac{d}{dx}(e^{-x})\\right)
$$<\/p>\n\n\n\n

$$
= \\frac{1}{2}(e^x + e^{-x})
$$<\/p>\n\n\n\n

This is the definition of $\\cosh(x)$:<\/p>\n\n\n\n

$$
= \\cosh(x)
$$<\/p>\n\n\n\n

(2) Hyperbolic Cosine $cosh$:<\/strong>
$$
\\frac{d}{dx} \\cosh(x) = \\sinh(x)
$$
The derivative of hyperbolic cosine is hyperbolic sine.<\/p>\n\n\n\n

Proof<\/strong>
The hyperbolic sine and cosine functions are defined as follows:<\/p>\n\n\n\n

$$
\\sinh(x) = \\frac{e^x – e^{-x}}{2}
$$
$$
\\cosh(x) = \\frac{e^x + e^{-x}}{2}
$$<\/p>\n\n\n\n

Now,<\/p>\n\n\n\n

$$
\\frac{d}{dx}\\cosh(x) = \\frac{d}{dx}\\left( \\frac{e^x + e^{-x}}{2}\\right)
$$<\/p>\n\n\n\n

$$
= \\frac{1}{2}\\left( \\frac{d}{dx}(e^x) + \\frac{d}{dx}(e^{-x})\\right)
$$<\/p>\n\n\n\n

$$
= \\frac{1}{2}(e^x – e^{-x})
$$<\/p>\n\n\n\n

This is the definition of $\\sinh(x)$:<\/p>\n\n\n\n

$$
= \\sinh(x)
$$<\/p>\n\n\n\n

(3) Hyperbolic Tangent $tanh$:<\/strong>
$$
\\frac{d}{dx} \\tanh(x))= sech^2(x)
$$
The derivative of hyperbolic tangent is the square of hyperbolic secant.<\/p>\n\n\n\n

Proof<\/strong>
$$
\\frac{d}{dx}\\tanh(x) = \\frac{d}{dx}\\left( \\frac{\\sinh(x)}{\\cosh(x)}\\right) = \\frac{\\cosh(x) \\frac{d}{dx} \\sinh(x)) – \\sinh(x)\\frac{d}{dx} \\cosh(x))}{\\cosh^2(x)}
$$<\/p>\n\n\n\n

We know that $\\frac{d}{dx} \\sinh(x) = \\cosh(x)$ and $\\frac{d}{dx} \\cosh(x) = \\sinh(x)$, so substituting these in gives:<\/p>\n\n\n\n

$$
= \\frac{\\cosh(x)\\cosh(x) – \\sinh(x)\\sinh(x)}{\\cosh^2(x)}
$$<\/p>\n\n\n\n

Simplifying the numerator:<\/p>\n\n\n\n

$$
= \\frac{\\cosh^2(x) – \\sinh^2(x)}{\\cosh^2(x)}
$$<\/p>\n\n\n\n

Using the identity \\$cosh^2(x) – \\sinh^2(x) = 1$, we get:<\/p>\n\n\n\n

$$
= \\frac{1}{\\cosh^2(x)}
$$<\/p>\n\n\n\n

$$
\\frac{d}{dx} \\tanh(x)) = sech^2(x)
$$<\/p>\n\n\n\n

(4) Hyperbolic Cotangent $coth$:<\/strong>
$$
\\frac{d}{dx} \\coth(x) = -csch^2(x)
$$
The derivative of hyperbolic cotangent is the negative square of hyperbolic cosecant.
Proof<\/strong><\/p>\n\n\n\n

$$
\\frac{d}{dx} \\coth(x)= \\frac{d}{dx}\\left( \\frac{\\cosh(x)}{\\sinh(x)}\\right) = \\frac{\\sinh(x)\\frac{d}{dx} \\cosh(x)) – \\cosh(x)\\frac{d}{dx} \\sinh(x))}{\\sinh^2(x)}
$$<\/p>\n\n\n\n

We know that $\\frac{d}{dx} \\cosh(x)) = \\sinh(x)$ and $\\frac{d}{dx}\\sinh(x)) = \\cosh(x)$, so substituting these in gives:<\/p>\n\n\n\n

$$
= \\frac{\\sinh(x)\\sinh(x) – \\cosh(x)\\cosh(x)}{\\sinh^2(x)}
$$<\/p>\n\n\n\n

$$
= \\frac{\\sinh^2(x) – \\cosh^2(x)}{\\sinh^2(x)}
$$<\/p>\n\n\n\n

Using the identity $\\cosh^2(x) – \\sinh^2(x) = 1$, we can rewrite the numerator as -1<\/p>\n\n\n\n

$$
= \\frac{-1}{\\sinh^2(x)}
$$<\/p>\n\n\n\n

$$
\\frac{d}{dx}$coth(x)) = -csch^2(x)
$$<\/p>\n\n\n\n

This completes the proof.<\/p>\n\n\n\n

(5) Hyperbolic Secant $sech$:<\/strong>
$$
\\frac{d}{dx}(sech(x) = -sech(x)\\tanh(x)
$$
The derivative of hyperbolic secant involves both hyperbolic secant and hyperbolic tangent.<\/p>\n\n\n\n

Proof<\/strong><\/p>\n\n\n\n

$$
\\frac{d}{dx} sech(x) = \\frac{d}{dx}\\left( \\frac{1}{\\cosh(x)}\\right) = \\frac{\\cosh(x)(0) – 1 \\sinh(x))}{\\cosh^2(x)}
$$<\/p>\n\n\n\n

$$
= \\frac{-\\sinh(x)}{\\cosh^2(x)}
$$<\/p>\n\n\n\n

Now, we can express $\\sinh(x)$ in terms of $\\tanh(x)$ and $\\cosh(x)$. <\/p>\n\n\n\n

Recall that $\\tanh(x) = \\frac{\\sinh(x)}{\\cosh(x)}$, so $\\sinh(x) = \\tanh(x) \\cdot \\cosh(x)$. <\/p>\n\n\n\n

Substituting this into our equation:<\/p>\n\n\n\n

$$
= \\frac{-\\tanh(x) \\cdot \\cosh(x)}{\\cosh^2(x)}
$$<\/p>\n\n\n\n

$$
= -\\tanh(x) \\cdot \\frac{1}{\\cosh(x)}
$$<\/p>\n\n\n\n

Since $\\frac{1}{\\cosh(x)}$ is $sech(x)$, we finally have:<\/p>\n\n\n\n

$$
\\frac{d}{dx}(sech(x)) = -sech(x)\\tanh(x)
$$<\/p>\n\n\n\n

(6) Hyperbolic Cosecant $csch$:<\/strong>
$$
\\frac{d}{dx}(csch(x) = -csch(x)\\coth(x)
$$
The derivative of hyperbolic cosecant involves both hyperbolic cosecant and hyperbolic cotangent.<\/p>\n\n\n\n

Proof<\/strong><\/p>\n\n\n\n

$$
\\frac{d}{dx} csch(x) = \\frac{d}{dx}\\left( \\frac{1}{\\sinh(x)}\\right) = \\frac{\\sinh(x)(0) – 1 \\cosh(x))}{\\sinh^2(x)}
$$<\/p>\n\n\n\n

$$
= \\frac{-\\cosh(x)}{\\sinh^2(x)}
$$<\/p>\n\n\n\n

Now, we can express $\\cosh(x)$ in terms of $\\coth(x)$ and $\\sinh(x)$. <\/p>\n\n\n\n

Recall that $\\coth(x) = \\frac{\\cosh(x)}{\\sinh(x)}$, so $\\cosh(x) = \\coth(x) \\cdot \\sinh(x)$. <\/p>\n\n\n\n

Substituting this into our equation:<\/p>\n\n\n\n

$$
= \\frac{-\\coth(x) \\cdot \\sinh(x)}{\\sinh^2(x)}
$$<\/p>\n\n\n\n

$$
= -\\coth(x) \\cdot \\frac{1}{\\sinh(x)}
$$<\/p>\n\n\n\n

Since $\\frac{1}{\\sinh(x)}$ is $csch(x)$, we finally have:<\/p>\n\n\n\n

$$
\\frac{d}{dx}(csch(x)) = -csch(x)\\coth(x)
$$<\/p>\n","protected":false},"excerpt":{"rendered":"

The derivatives of the hyperbolic functions are quite straightforward and somewhat analogous to the derivatives of their trigonometric counterparts. Here are the derivatives for the six primary hyperbolic functions: (1) Hyperbolic Sine $sinh$:$$\\frac{d}{dx} \\sinh(x) = \\cosh(x)$$The derivative of hyperbolic sine is hyperbolic cosine. ProofThe hyperbolic sine and cosine functions are defined as follows: $$\\sinh(x) = […]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[498],"tags":[],"class_list":["post-8648","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"\nDerivatives of hyperbolic functions - physicscatalyst's Blog<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/physicscatalyst.com\/article\/derivatives-of-hyperbolic-functions\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Derivatives of hyperbolic functions - physicscatalyst's Blog\" \/>\n<meta property=\"og:description\" content=\"The derivatives of the hyperbolic functions are quite straightforward and somewhat analogous to the derivatives of their trigonometric counterparts. 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Here are the derivatives for the six primary hyperbolic functions: (1) Hyperbolic Sine $sinh$:$$frac{d}{dx} sinh(x) = cosh(x)$$The derivative of hyperbolic sine is hyperbolic cosine. 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Here are the derivatives for the six primary hyperbolic functions: (1) Hyperbolic Sine $sinh$:$$\\frac{d}{dx} \\sinh(x) = \\cosh(x)$$The derivative of hyperbolic sine is hyperbolic cosine. ProofThe hyperbolic sine and cosine functions are defined as follows: $$\\sinh(x) =…","_links":{"self":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/8648","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/comments?post=8648"}],"version-history":[{"count":6,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/8648\/revisions"}],"predecessor-version":[{"id":8654,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/8648\/revisions\/8654"}],"wp:attachment":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/media?parent=8648"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/categories?post=8648"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/tags?post=8648"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}