Q.
Find the derivatives of all of these (1 to 15)
\( \log(\tanh x) \)
➤
\begin{aligned}
f(x) & =\log(\tanh x) \\
f'(x) & = \frac{d(\log(\tanh x))}{dx} \\
& = \frac{1}{\tanh x} \frac{d(\tanh x)}{dx} \\
& = \frac{1}{\tanh x} \frac{d(\sinh x)}{dx} \\
& = \frac{1}{\tanh x} \cosh x \\
& = \frac{\cosh x}{\sinh x} \\
& = \coth x
\end{aligned}
Q.
\( \log(\sinh \frac{x}{a})\)
➤
\begin{aligned}
f(x) & = \log(\sinh \frac{x}{a}) \\
f'(x) & = \frac{d(\log(\sinh \frac{x}{a}))}{dx} \\
& = \frac{1}{\sinh \frac{x}{a}} \frac{d(\sinh \frac{x}{a})}{dx} \\
& = \frac{1}{\sinh \frac{x}{a}} \cosh \frac{x}{a} \\
& = \coth \frac{x}{a}
\end{aligned}
Q.
\(e^{\sinh x}\)
➤
\begin{aligned}
f(x) & = e^{\sinh x} \\
f'(x) & = \frac{d(e^{\sinh x})}{dx} \\
& = e^{\sinh x} \frac{d(\sinh x)}{dx} \\
& = e^{\sinh x} \cosh x \\
& = \cosh x e^{\sinh x}
\end{aligned}
Q.
\(e^{\cosh^{-1} \frac{x}{a}}\)
➤
\begin{aligned}
f(x) & = e^{\cosh^{-1} \frac{x}{a}} \\
f'(x) & = \frac{d(e^{\cosh^{-1} \frac{x}{a}})}{dx} \\
& = e^{\cosh^{-1} \frac{x}{a}} \frac{d(\cosh^{-1} \frac{x}{a})}{dx} \\
& = e^{\cosh^{-1} \frac{x}{a}} \frac{1}{\sqrt{\frac{x^2}{a^2}-1}} \frac{d(\frac{x}{a})}{dx} \\
& = \frac{e^{\cosh^{-1} \frac{x}{a}}}{\frac{\sqrt{{x^2}-{a^2}}}{a}} \frac{1}{a} \\
& = \frac{e^{\cosh^{-1} \frac{x}{a}}}{\sqrt{{x^2}-{a^2}}} \\
\end{aligned}
Q.
\(\sech(\tan^{-1} x)\)
➤
\begin{aligned}
f(x) & = \sech(\tan^{-1} x) \\
f'(x) & = \frac{d(\sech(\tan^{-1} x))}{dx} \\
& = \frac{d(\sech(\tan^{-1} x))}{d(\tan^{-1} x)} \frac{d(\tan^{-1} x)}{dx} \\
& = - {\sech (tan^{-1} x) \cdot \tanh (\tanh ^ {-1}x)} \frac{1}{1+x^2} \\
\end{aligned}
Q.
\(\sech^{-1}x-\cosh^{-1}x\)
➤
\begin{aligned}
f(x) & = \sech^{-1}x-\cosh^{-1}x \\
f'(x) & = \frac{d(\sech^{-1}x-\cosh^{-1}x)}{dx} \\
& = \frac{d(\sech^{-1}x)}{dx}-\frac{d(\cosh^{-1}x)}{dx} \\
& = -\frac{1}{x\sqrt{x^2-1}}-\frac{1}{\sqrt{x^2-1}} \\
& = -\frac{1}{\sqrt{x^2-1}}(1+\frac{1}{x}) \\
& = -\frac{1}{\sqrt{x^2-1}}\frac{x+1}{x} \\
& = -\frac{x+1}{x\sqrt{x^2-1}}
\end{aligned}
Q.
\(\arctan(\sinh x)\)
➤
\begin{aligned}
f(x) & = \arctan(\sinh x) \\
f'(x) & = \frac{d(\arctan(\sinh x))}{dx} \\
& = \frac{d(\arctan(\sinh x))}{d(\sinh x)} \frac{d(\sinh x)}{dx} \\
& = \frac{1}{1+(\sinh x)^2} \cosh x \\
& = \frac{\cosh x}{1+\sinh^2 x} \\
& = \frac{\cosh x}{\cosh^2 x} \\
& = \sech x
\end{aligned}
Q.
\(2 \tanh^{-1} (\tan {\frac{1}{2}x})\)
➤
\begin{aligned}
f(x) & = 2 \tanh^{-1} (\tan {\frac{x}{2}}) \\
f'(x) & = \frac{d(2 \tanh^{-1} (\tan {\frac{x}{2}}))}{dx} \\
& = 2 \frac{d(\tanh^{-1} (\tan {\frac{x}{2}}))}{d(\tan {\frac{x}{2}})} \frac{d(\tan {\frac{x}{2}})}{dx} \\
& = 2 \frac{1}{1-(\tan {\frac{x}{2}})^2} \frac{1}{2} \sec^2 {\frac{x}{2}} \\
& = \frac{\sec^2 {\frac{x}{2}} }{1-\tan^2 {\frac{x}{2}}} \\
& = \frac{\frac{1}{\cos^2 \frac{x}{2}}}{1 - \frac{sin^2 \frac{x}{2}}{cos^2 \frac{x}{2}}} \\
& = \frac{\frac{1}{\cos^2 \frac{x}{2}}}{\frac{cos^2 \frac{x}{2} -sin^2 \frac{x}{2}}{cos^2 \frac{x}{2}}} \\
& = \frac{1}{cos{2\frac{x}{2}}} \\
& = \sec x \\ \\
\end{aligned}
Q.
\(x^{\cosh \frac{x}{a}}\)
➤
\begin{aligned}
f(x) & = x^{\cosh \frac{x}{a}} \\
f'(x) & = \frac{d(x^{\cosh \frac{x}{a}})}{dx} \\
& = \frac{d(e^{log {x^{\cosh \frac{x}{a}}}})}{dx} \\
& = \frac{d(e^{\cosh \frac{x}{a} log x})}{dx} \\
& = e^{\cosh \frac{x}{a} log x} \frac{d(\cosh \frac{x}{a} log x)}{dx} \\
& = x^{\cosh \frac{x}{a}} {(\frac{\sinh \frac{x}{a} \log x }{a}+ {(\cosh \frac{x}{a})}\frac{1}{x} )} \\
\end{aligned}
Q.
\(x^{\sinh {\frac{x^2}{a}}}\)
➤
\begin{aligned}
f(x) & = x^{\sinh {\frac{x^2}{a}}} \\
f'(x) & = \frac{d(x^{\sinh {\frac{x^2}{a}}})}{dx} \\
& = \frac{d(e^{log {x^{\sinh {\frac{x^2}{a}}}}})}{dx} \\
& = \frac{d(e^{\sinh {\frac{x^2}{a}} log x})}{dx} \\
& = e^{\sinh {\frac{x^2}{a}} log x} \frac{d(\sinh {\frac{x^2}{a}} log x)}{dx} \\
& = x^{\sinh {\frac{x^2}{a}}} {(\frac{2x \cosh {\frac{x^2}{a}} \log x }{a}+ {(\sinh {\frac{x^2}{a}})}\frac{1}{x} )} \\
\end{aligned}
Q.
\(x^{\cosh^2 {\frac{x}{a}}}\)
➤
\begin{aligned}
f(x) & = x^{\cosh^2 {\frac{x}{a}}} \\
f'(x) & = \frac{d(x^{\cosh^2 {\frac{x}{a}}})}{dx} \\
& = \frac{d(e^{log {x^{\cosh^2 {\frac{x}{a}}}}})}{dx} \\
& = \frac{d(e^{\cosh^2 {\frac{x}{a}} log x})}{dx} \\
& = e^{\cosh^2 {\frac{x}{a}} log x} \frac{d(\cosh^2 {\frac{x}{a}} log x)}{dx} \\
& = x^{\cosh^2 {\frac{x}{a}}} {(\frac{2 \sinh {\frac{x}{a}} \cosh {\frac{x}{a}} \log x }{a}+ {(\cosh^2 {\frac{x}{a}})}\frac{1}{x} )} \\
\end{aligned}
Q.
\((\sinh \frac{x}{a})^{x^2}\)
➤
\begin{aligned}
f(x) & = (\sinh \frac{x}{a})^{x^2} \\
f'(x) & = \frac{d((\sinh \frac{x}{a})^{x^2})}{dx} \\
& = \frac{d(e^{log {(\sinh \frac{x}{a})^{x^2}}})}{dx} \\
& = \frac{d(e^{x^2 \log {\sinh \frac{x}{a}}})}{dx} \\
& = e^{x^2 \log {\sinh \frac{x}{a}}} \frac{d(x^2 \log {\sinh \frac{x}{a}})}{dx} \\
& = (\sinh \frac{x}{a})^{x^2} {( 2x \log {\sinh \frac{x}{a}} + {x^2}\frac{1}{\sinh \frac{x}{a}}\cosh \frac{x}{a}\frac{1}{a} )} \\
& = x(\sinh \frac{x}{a})^{x^2} {( 2 \log {\sinh \frac{x}{a}} + \frac{{x}\coth \frac{x}{a}}{a} )} \\
\end{aligned}
Q.
\((\cosh \frac{x}{a})^{log x}\)
➤
\begin{aligned}
f(x) & = (\cosh \frac{x}{a})^{log x} \\
f'(x) & = \frac{d((\cosh \frac{x}{a})^{log x})}{dx} \\
& = \frac{d(e^{log {(\cosh \frac{x}{a})^{log x}}})}{dx} \\
& = \frac{d(e^{log x \log {\cosh \frac{x}{a}}})}{dx} \\
& = e^{log x \log {\cosh \frac{x}{a}}} \frac{d(log x \log {\cosh \frac{x}{a}})}{dx} \\
& = (\cosh \frac{x}{a})^{log x} {( \frac{1}{x} \log {\cosh \frac{x}{a}} + {log x}\frac{1}{\cosh \frac{x}{a}}\sinh \frac{x}{a}\frac{1}{a} )} \\
& = \frac{log x}{x}(\cosh \frac{x}{a})^{log x} {( \log {\cosh \frac{x}{a}} + \frac{{log x}\tanh \frac{x}{a}}{a} )} \\
\end{aligned}
Q.
\((\cosh x)^{\sinh^{-1}x}\)
➤
\begin{aligned}
f(x) & = (\cosh x)^{\sinh^{-1}x} \\
f'(x) & = \frac{d((\cosh x)^{\sinh^{-1}x})}{dx} \\
& = \frac{d(e^{log {(\cosh x)^{\sinh^{-1}x}}})}{dx} \\
& = \frac{d(e^{\sinh^{-1}x \log {\cosh x}})}{dx} \\
& = e^{\sinh^{-1}x \log {\cosh x}} \frac{d(\sinh^{-1}x \log {\cosh x})}{dx} \\
& = (\cosh x)^{\sinh^{-1}x} {( \frac{1}{\sqrt{1+x^2}} \log {\cosh x} + {\sinh^{-1}x}\frac{1}{\cosh x}\sinh x )} \\
& = (\cosh x)^{\sinh^{-1}x} {( \frac{1}{\sqrt{1+x^2}} \log {\cosh x} + {\sinh^{-1}x \tanh x} )} \\
\end{aligned}
Q.
\((\sinh \frac{x}{a}+\cosh \frac{x}{a})^{nx}\)
➤
\begin{aligned}
f(x) & = (\sinh \frac{x}{a}+\cosh \frac{x}{a})^{nx} \\
f'(x) & = \frac{d((\sinh \frac{x}{a}+\cosh \frac{x}{a})^{nx})}{dx} \\
& = \frac{d(e^{log {(\sinh \frac{x}{a}+\cosh \frac{x}{a})^{nx}}})}{dx} \\
& = \frac{d(e^{nx \log {(\sinh \frac{x}{a}+\cosh \frac{x}{a})}})}{dx} \\
& = e^{nx \log {(\sinh \frac{x}{a}+\cosh \frac{x}{a})} } \frac{d(nx \log {(\sinh \frac{x}{a}+\cosh \frac{x}{a})})}{dx} \\
& = (\sinh \frac{x}{a}+\cosh \frac{x}{a})^{nx} {( n \log {(\sinh \frac{x}{a}+\cosh \frac{x}{a})} + {nx}\frac{1}{\sinh \frac{x}{a}+\cosh \frac{x}{a}}(\cosh \frac{x}{a}+\sinh \frac{x}{a})\frac{1}{a} )} \\
& = n(\sinh \frac{x}{a}+\cosh \frac{x}{a})^{nx} {[ \log {(\sinh \frac{x}{a}+\cosh \frac{x}{a})} + \frac{x}{a} ]} \\
\end{aligned}
