Here we will discuss the derivative of hyperbolic functions:
[katex]\boxed{\frac d{dx}\left(\sin hx\right)=\cos hx}[/katex]
[katex]\boxed{\frac d{dx}\left(\cos hx\right)=\sin hx}[/katex]
[katex]\boxed{\frac d{dx}\left(\tan hx\right)=sec^2hx}[/katex]
[katex]\boxed{\frac d{dx}\left(cothx\right)=-cosec^2hx}[/katex]
[katex]\boxed{\frac d{dx}\left(sechx\right)=-sechx.\tan hx}[/katex]
[katex]\boxed{\frac d{dx}\left(cosechx\right)=-cosechx.cothx}[/katex]
Derivative of sin hyperbolic functions:
[katex]\frac d{dx}\left(\sin hx\right)=\cos hx[/katex]
[katex]\sin hx=\left(\frac{e^x-e^{-x}}2\right)[/katex]
differentiating w.r.t x
[katex]\frac d{dx}\sin hx=\frac d{dx}\left(\frac{e^x-e^{-x}}2\right)[/katex]
[katex]\frac d{dx}\sin hx=\frac12\frac d{dx}\left(e^x-e^{-x}\right)[/katex]
Now by using the sum and difference rule.
[katex]\frac d{dx}\sin hx=\frac12\left[\frac d{dx}e^x-\frac d{dx}e^{-x}\right][/katex]
[katex]\frac d{dx}\sin hx=\frac12\left[e^x-e^{-x}\left(-1\right)\right][/katex]
[katex]\frac d{dx}\sin hx=\frac12\left(e^x+e^{-x}\right)[/katex]
[katex]\boxed{\frac d{dx}\sin hx=\frac{e^x+e^{-x}}2}[/katex]
[katex]\boxed{\frac d{dx}\sin hx=\cos hx}[/katex]
This is a required derivative of sinhx.
Derivative of cos hyperbolic functions:
[katex]\frac d{dx}\left(\cos hx\right)=\sin hx[/katex]
[katex]\cos hx=\left(\frac{e^x+e^{-x}}2\right)[/katex]
differentiating w.r.t x
[katex]\frac d{dx}\sin hx=\frac d{dx}\left(\frac{e^x+e^{-x}}2\right)[/katex]
[katex]\frac d{dx}\sin hx=\frac12\frac d{dx}\left(e^x+e^{-x}\right)[/katex]
Now by using the sum and difference rule.
[katex]\frac d{dx}\cos hx=\frac12\left[\frac d{dx}e^x+\frac d{dx}e^{-x}\right][/katex]
[katex]\frac d{dx}\cos hx=\frac12\left[e^x+e^{-x}\left(-1\right)\right][/katex]
[katex]\frac d{dx}\cos hx=\frac12\left(e^x-e^{-x}\right)[/katex]
[katex]\boxed{\frac d{dx}\cos hx=\frac{e^x-e^{-x}}2}[/katex]
[katex]\boxed{\frac d{dx}\cos hx=\sin hx}[/katex]
This is required derivative of coshx.
Derivative of Tanhx:
[katex]\boxed{\frac d{dx}\left(\tan hx\right)=sec^2hx}[/katex]
[katex]Tanhx=\frac{\sin hx}{\cos hx}[/katex]
[katex]\frac{\sin hx}{\cos hx}=\frac{\displaystyle\frac{e^x-e^{-x}}2}{\displaystyle\frac{e^x+e^{-x}}2}[/katex]
[katex]\frac{\sin hx}{\cos hx}==\frac{e^x-e^{-x}}2\times\frac2{e^x+e^{-x}}[/katex]
[katex]Tanhx=\frac{e^x-e^{-x}}{e^x+e^{-x}}[/katex]
differentiating w.r.t x
[katex]\frac d{dx}Tanhx=\frac d{dx}\left[\frac{e^x-e^{-x}}{e^x+e^{-x}}\right][/katex]
Now by using the quotient rule.
[katex]\frac d{dx}Tanhx=\frac{\left(e^x+e^{-x}\right).{\displaystyle\frac d{dx}}\left(e^x-e^{-x}\right)-\left(e^x-e^{-x}\right).{\displaystyle\frac d{dx}}\left(e^x+e^{-x}\right)}{\left(e^x+e^{-x}\right)^2}[/katex]
[katex]\frac d{dx}Tanhx=\frac{\left(e^x+e^{-x}\right).\left(e^x-e^{-x}\times-1\right)-\left(e^x-e^{-x}\right).\left(e^x+e^{-x}\times-1\right)}{\left(e^x+e^{-x}\right)^2}[/katex]
[katex]\frac d{dx}Tanhx=\frac{\left(e^x+e^{-x}\right).\left(e^x+e^{-x}\right)-\left(e^x-e^{-x}\right).\left(e^x-e^{-x}\right)}{\left(e^x+e^{-x}\right)^2}[/katex]
[katex]\frac d{dx}Tanhx=\frac{\left(e^x+e^{-x}\right)^2-\left(e^x-e^{-x}\right)^2}{\left(e^x+e^{-x}\right)^2}[/katex]
[katex]\frac d{dx}Tanhx=\frac{\left(e^{2x}+e^{-2x}+2\right)-\left(e^{2x}-e^{-2x}-2\right)}{\left(e^x+e^{-x}\right)^2}[/katex]
[katex]\frac d{dx}Tanhx=\frac{e^{2x}+e^{-2x}+2-e^{2x}-e^{-2x}+2}{\left(e^x+e^{-x}\right)^2}[/katex]
[katex]\frac d{dx}Tanhx=\frac4{\left(e^x+e^{-x}\right)^2}[/katex]
[katex]\frac d{dx}Tanhx=\left(\frac2{e^x+e^{-x}}\right)^2[/katex]
[katex]\boxed{\frac d{dx}Tanhx=sech^2x}[/katex]
This is required derivative of Tanhx.
Derivative of cosec hyperbolic functions:
[katex]\boxed{\frac d{dx}\left(cosechx\right)=-cosechx.cothx}[/katex]
As we know that
[katex]\cos echx=\frac2{e^x-e^{-x}}[/katex]
[katex]\frac d{dx}\cos echx=\frac d{dx}\left(\frac2{e^x-e^{-x}}\right)[/katex]
[katex]\frac d{dx}\cos echx=2\frac d{dx}\left(\frac1{e^x-e^{-x}}\right)[/katex]
[katex]\frac d{dx}\cos echx=2\frac d{dx}\left(e^x-e^{-x}\right)^{-1}[/katex]
Now by using the power rule:
[katex]\frac d{dx}\cos echx=2.(-1)\left(e^x-e^{-x}\right)^{-2}.\frac d{dx}(e^x-e^{-x})[/katex]
[katex]\frac d{dx}\cos echx=2.(-1)\left(e^x-e^{-x}\right)^{-2}.(e^x-e^{-x}\times-1)[/katex]
[katex]\frac d{dx}\cos echx=2.(-1)\left(e^x-e^{-x}\right)^{-2}.(e^x+e^{-x})[/katex]
[katex]\frac d{dx}\cos echx=-2.\frac{(e^x+e^{-x})}{\left(e^x-e^{-x}\right)^2}[/katex]
[katex]\frac d{dx}\cos echx=-\frac2{\left(e^x-e^{-x}\right)}.\frac{(e^x+e^{-x})}{\left(e^x-e^{-x}\right)}[/katex]
[katex]\boxed{\frac d{dx}\cos echx=-\cos echx.cothx}[/katex]
This is a required derivative of cosechx.
Derivative of sec hyperbolic functions:
[katex]\boxed{\frac d{dx}\left(sechx\right)=-sechx.tanhx}[/katex]
As we know that:
[katex]sechx=\frac2{e^x+e^{-x}}[/katex]
[katex]\frac d{dx}sechx=\frac d{dx}\left(\frac2{e^x+e^{-x}}\right)[/katex]
[katex]\frac d{dx}sechx=2\frac d{dx}\left(\frac1{e^x+e^{-x}}\right)[/katex]
[katex]\frac d{dx}sechx=2\frac d{dx}\left(e^x+e^{-x}\right)^{-1}[/katex]
Now by using power rule:
[katex]\frac d{dx}sechx=2.(-1)\left(e^x+e^{-x}\right)^{-2}.\frac d{dx}(e^x+e^{-x})[/katex]
[katex]\frac d{dx}sechx=2.(-1)\left(e^x+e^{-x}\right)^{-2}.(e^x+e^{-x}\times-1)[/katex]
[katex]\frac d{dx}secchx=2.(-1)\left(e^x+e^{-x}\right)^{-2}.(e^x-e^{-x})[/katex]
[katex]\frac d{dx}sechx=-2.\frac{(e^x+e^{-x})}{\left(e^x-e^{-x}\right)^2}[/katex]
[katex]\frac d{dx}sechx=-\frac2{\left(e^x+e^{-x}\right)}.\frac{(e^x+e^{-x})}{\left(e^x-e^{-x}\right)}[/katex]
[katex]\boxed{\frac d{dx}\cos echx=-\cos echx.cothx}[/katex]
This is required derivative of sechx.
Derivative of cot hyperbolic functions:
[katex]\boxed{\frac d{dx}\left(\cot hx\right)=cosec^2hx}[/katex]
[katex]cothx=\frac{\cos hx}{\sin hx}[/katex]
[katex]\frac{\cos hx}{\sin hx}=\frac{\displaystyle\frac{e^x+e^{-x}}2}{\displaystyle\frac{e^x-e^{-x}}2}[/katex]
[katex]\frac{\cos hx}{\sin hx}==\frac{e^x+e^{-x}}2\times\frac2{e^x-e^{-x}}[/katex]
[katex]cothx=\frac{e^x+e^{-x}}{e^x-e^{-x}}[/katex]
differentiating w.r.t x
[katex]\frac d{dx}cothx=\frac d{dx}\left[\frac{e^x+e^{-x}}{e^x-e^{-x}}\right][/katex]
[katex]\frac d{dx}cothx=\frac{\left(e^x-e^{-x}\right).{\displaystyle\frac d{dx}}\left(e^x+e^{-x}\right)-\left(e^x+e^{-x}\right).{\displaystyle\frac d{dx}}\left(e^x-e^{-x}\right)}{\left(e^x-e^{-x}\right)^2}[/katex]
[katex]\frac d{dx}cothx=\frac{\left(e^x-e^{-x}\right).\left(e^x+e^{-x}\times-1\right)-\left(e^x+e^{-x}\right).\left(e^x-e^{-x}\times-1\right)}{\left(e^x-e^{-x}\right)^2}[/katex]
[katex]\frac d{dx}cothx=\frac{\left(e^x-e^{-x}\right).\left(e^x-e^{-x}\right)-\left(e^x+e^{-x}\right).\left(e^x+e^{-x}\right)}{\left(e^x-e^{-x}\right)^2}[/katex]
[katex]\frac d{dx}cothx=\frac{\left(e^x-e^{-x}\right)^2-\left(e^x+e^{-x}\right)^2}{\left(e^x-e^{-x}\right)^2}[/katex]
[katex]\frac d{dx}cothx=\frac{\left(e^{2x}+e^{-2x}-2\right)-\left(e^{2x}+e^{-2x}+2\right)}{\left(e^x-e^{-x}\right)^2}[/katex]
[katex]\frac d{dx}cothx=\frac{e^{2x}+e^{-2x}-2-e^{2x}-e^{-2x}-2}{\left(e^x-e^{-x}\right)^2}[/katex]
[katex]\frac d{dx}cothx=\frac{-4}{\left(e^x-e^{-x}\right)^2}[/katex]
[katex]\frac d{dx}cothx=-\left(\frac2{e^x+e^{-x}}\right)^2[/katex]
[katex]\boxed{\frac d{dx}cothx=-cosech^2x}[/katex]