Question

Prove the formulas given in Table 1 for the derivatives of the functions (a) cosh (b) tanh (c) csch (d) sech and (e) coth

Short answer

(a) It is proved that derivative of \(\cosh x\) is \(\sinh x\).

(b) It is proved that derivative of \(\tanh x\) is \({{\mathop{\rm sech}\nolimits} ^2}x\).

(c) It is proved that derivative of \({\mathop{\rm csch}\nolimits} x\) is \( - {\mathop{\rm csch}\nolimits} x\coth x\).

(d) It is proved that derivative of \({\mathop{\rm sech}\nolimits} x\) is \( - {\mathop{\rm sech}\nolimits} x\tanh x\).

(e) It is proved that derivative of \(\coth x\) is \( - {{\mathop{\rm csch}\nolimits} ^2}x\).

Step-by-step solution

Step 1:Find the answer for part (a)

Find the derivative of the function \(\cosh x\).

\(\begin{aligned}\frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\cosh x} \right) &= \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\frac{1}{2}\left( {{e^x} + {e^{ - x}}} \right)} \right)\\ &= \frac{1}{2}\left( {{e^x} - {e^{ - x}}} \right)\\ &= \sinh x\end{aligned}\)

It is proved that derivative of \(\cosh x\) is \(\sinh x\).

 Step 2: Find the answer for part (b)

Find the derivative of the function \(\tanh x\).

\(\begin{aligned}\frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\tanh x} \right) &= \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\frac{{\sinh x}}{{\cosh x}}} \right)\\ &= \frac{{\cosh x\frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\sinh x} \right) - \sinh x\frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\cosh x} \right)}}{{{{\cosh }^2}x}}\\ &= \frac{{{{\cosh }^2}x - {{\sinh }^2}x}}{{{{\cosh }^2}x}}\\ &= \frac{1}{{{{\cosh }^2}x}}\\ &= {{\mathop{\rm sech}\nolimits} ^2}x\end{aligned}\)

It is proved that derivative of \(\tanh x\) is \({{\mathop{\rm sech}\nolimits} ^2}x\).

Find the answer for part (c)

Find the derivative of the function \({\mathop{\rm csch}\nolimits} x\).

\(\begin{aligned}\frac{{\rm{d}}}{{{\rm{d}}x}}\left( {{\mathop{\rm csch}\nolimits} x} \right) &= \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\frac{1}{{\sinh x}}} \right)\\ &= \frac{{0 - 1\left( {\cosh x} \right)}}{{{{\sinh }^2}x}}\\ &= - {\mathop{\rm csch}\nolimits} x\coth x\end{aligned}\)

It is proved that derivative of \({\mathop{\rm csch}\nolimits} x\) is \( - {\mathop{\rm csch}\nolimits} x\coth x\).

Find the answer for part (d)

Find the derivative of \({\mathop{\rm sech}\nolimits} x\).

\(\begin{aligned}\frac{{\rm{d}}}{{{\rm{d}}x}}\left( {{\mathop{\rm sech}\nolimits} x} \right) &= \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\frac{1}{{\cosh x}}} \right)\\ &= \frac{{ - \sinh x}}{{{{\cosh }^2}x}}\\ &= - {\mathop{\rm sech}\nolimits} x\tanh x\end{aligned}\)

It is proved that derivative of \({\mathop{\rm sech}\nolimits} x\) is \( - {\mathop{\rm sech}\nolimits} x\tanh x\).

Find the answer for part (e)

Find the derivative of \(\coth x\).

\(\begin{aligned}\frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\coth x} \right) &= \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\frac{{\cosh x}}{{\sinh x}}} \right)\\ &= \frac{{\sinh x\left( {\sinh x} \right) - \cosh x\left( {\cosh x} \right)}}{{{{\sinh }^2}x}}\\&= \frac{{{{\sinh }^2}x - {{\cosh }^2}x}}{{{{\sinh }^2}x}}\\ &= - \frac{1}{{{{\sinh }^2}x}}\\ &= - {{\mathop{\rm csch}\nolimits} ^2}x\end{aligned}\)

It is proved that derivative of \(\coth x\) is \( - {{\mathop{\rm csch}\nolimits} ^2}x\).