ABSTRACT
A list of standard derivatives for hyperbolic functions is shown in the table below.
y or f(x)
dy dx https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3240.tif"/> or f’(x)
sinh ax
a cosh ax
cosh ax
a sinh ax
tanh ax
a sech2 ax
sech ax
–a sech ax tanh ax
cosech ax
–a cosech ax coth ax
coth ax
–a cosech2 ax
Application: Differentiate the following with respect to x:
y = 4 sh 2x − 3 7 ch 3x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3241.tif"/>
y = 5 th x 2 − 2 coth 4x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3242.tif"/>
dy dx = 4 ( 2 cosh 2 x ) − 3 7 ( 3 sinh 3 x ) = 8 c o s h 2 x − 9 7 s i n h 3 x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3243.tif"/>
dy dx = 5 ( 1 2 sec h 2 x 2 ) − 2 ( − 4 cosech 2 4 x) = 5 2 sec h 2 x 2 + 8 cosech 2 4 x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3244.tif"/>
Application: Differentiate the following with respect to the variable:
y = 4 sin 3 t ch 4t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3245.tif"/>
y = ln ( sh 3θ ) − 4 ch 2 3 θ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3246.tif"/>
y = 4 sin 3 t ch 4t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3247.tif"/> (i.e. a product) dy dt = ( 4 sin 3 t ) ( 4 sh 4t ) + ( ch 4t ) ( 4 ) ( 3 cos 3 t ) = 16 sin 3 t sh 4t + 12 ch 4t cos 3t = 4 ( 4 s i n 3 t s h 4 t + 3 c o s 3 t c h 4 t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3248.tif"/>
y = ln ( sh 3 θ ) − 4 ch 2 3 θ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3249.tif"/> (i.e. a function of a function) dy dθ = ( 1 sh 3θ ) ( 3 ch 3θ ) − ( 4 ) ( 2 ch 3 θ ) ( 3 sh 3θ ) = 3 coth 3 θ − 24 ch 3θ sh 3θ = 3 ( coth 3 θ − 8 ch 3θ sh 3θ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3250.tif"/>
