(1)

\begin{align*} \left(\sin x\right)' & =\lim_{\Delta x\rightarrow0}\frac{\sin(x+\Delta x)-\sin x}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\frac{\sin x\cos\Delta x+\cos x\sin\Delta x-\sin x}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\left(\frac{\cos x\sin\Delta x}{\Delta x}+\frac{\sin x\left(\cos\Delta x-1\right)}{\Delta x}\right)\\ & =\lim_{\Delta x\rightarrow0}\left(\frac{\cos x\sin\Delta x}{\Delta x}-\frac{\sin x\sin^{2}\Delta x}{\Delta x\left(\cos\Delta x+1\right)}\right)\\ & =\lim_{\Delta x\rightarrow0}\frac{\sin\Delta x}{\Delta x}\left(\cos x-\frac{\sin x\sin\Delta x}{\cos\Delta x+1}\right)\\ & =\cos x \end{align*}

(1)-2

\begin{align*} \left(\sin x\right)' & =\lim_{\Delta x\rightarrow0}\frac{\sin(x+\Delta x)-\sin x}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\frac{1}{\Delta x}2\cos\frac{2x+\Delta x}{2}\sin\frac{\Delta x}{2}\\ & =\lim_{\Delta x\rightarrow0}\cos\left(x+\frac{\Delta x}{2}\right)\frac{\sin\frac{\Delta x}{2}}{\frac{\Delta x}{2}}\\ & =\cos x \end{align*}

(2)

\begin{align*} \left(\cos x\right)' & =\left(\sin\left(x+\frac{\pi}{2}\right)\right)'\\ & =\frac{d\sin\left(x+\frac{\pi}{2}\right)}{x}\\ & =\cos\left(x+\frac{\pi}{2}\right)\\ & =\sin\left(x+\pi\right)\\ & =-\sin x \end{align*}

(2)-2

\begin{align*} \left(\cos x\right)' & =\lim_{\Delta x\rightarrow0}\frac{\cos(x+\Delta x)-\cos x}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\frac{\cos x\cos\Delta x-\sin x\sin\Delta x-\cos x}{\Delta x}\\ & =-\lim_{\Delta x\rightarrow0}\left(\frac{\sin x\sin\Delta x}{\Delta x}-\frac{\cos x\left(\cos\Delta x-1\right)}{\Delta x}\right)\\ & =-\lim_{\Delta x\rightarrow0}\left(\frac{\sin x\sin\Delta x}{\Delta x}+\frac{\cos x\sin^{2}\Delta x}{\Delta x\left(\cos\Delta x+1\right)}\right)\\ & =-\lim_{\Delta x\rightarrow0}\frac{\sin\Delta x}{\Delta x}\left(\sin x+\frac{\cos x\sin\Delta x}{\cos\Delta x+1}\right)\\ & =-\sin x \end{align*}

(2)-3

\begin{align*} \left(\cos x\right)' & =\lim_{\Delta x\rightarrow0}\frac{\cos(x+\Delta x)-\cos x}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\frac{1}{\Delta x}\left(-2\sin\frac{2x+\Delta x}{2}\sin\frac{\Delta x}{2}\right)\\ & =-\lim_{\Delta x\rightarrow0}\sin\left(x+\frac{\Delta x}{2}\right)\frac{\sin\frac{\Delta x}{2}}{\frac{\Delta x}{2}}\\ & =-\sin x \end{align*}

(3)

\begin{align*} \left(\tan x\right)' & =\left(\frac{\sin x}{\cos x}\right)'\\ & =\frac{\left(\sin x\right)'\cos x-\sin x\left(\cos x\right)'}{\cos^{2}x}\\ & =\frac{\cos^{2}x+\sin^{2}x}{\cos^{2}x}\\ & =\frac{1}{\cos^{2}x} \end{align*}

(3)-2

\begin{align*} \left(\tan x\right)' & =\lim_{\Delta x\rightarrow0}\frac{1}{\Delta x}\left(\frac{\sin\left(x+\Delta x\right)}{\cos(x+\Delta x)}-\frac{\sin x}{\cos x}\right)\\ & =\lim_{\Delta x\rightarrow0}\frac{1}{\cos x\cos(x+\Delta x)\Delta x}\left(\sin\left(x+\Delta x\right)\cos x-\cos(x+\Delta x)\sin x\right)\\ & =\lim_{\Delta x\rightarrow0}\frac{\sin\Delta x}{\cos x\cos(x+\Delta x)\Delta x}\\ & =\frac{1}{\cos^{2}x} \end{align*}

(1)

\begin{align*} \left(\sinh x\right)' & =\lim_{\Delta x\rightarrow0}\frac{\sinh(x+\Delta x)-\sinh x}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\frac{\sinh x\cosh\Delta x+\cosh x\sinh\Delta x-\sinh x}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\left(\frac{\cosh x\sinh\Delta x}{\Delta x}+\frac{\sinh x\left(\cosh\Delta x-1\right)}{\Delta x}\right)\\ & =\lim_{\Delta x\rightarrow0}\left(\frac{\cosh x\sinh\Delta x}{\Delta x}+\frac{\sinh x\sinh^{2}\Delta x}{\Delta x\left(\cosh\Delta x+1\right)}\right)\\ & =\lim_{\Delta x\rightarrow0}\frac{\sinh\Delta x}{\Delta x}\left(\cosh x+\frac{\sinh x\sinh\Delta x}{\cosh\Delta x+1}\right)\\ & =\cosh x \end{align*}

(1)-2

\begin{align*} \left(\sinh x\right)' & =\lim_{\Delta x\rightarrow0}\frac{\sinh(x+\Delta x)-\sinh x}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\frac{1}{\Delta x}2\cosh\frac{2x+\Delta x}{2}\sinh\frac{\Delta x}{2}\\ & =\lim_{\Delta x\rightarrow0}\cosh\left(x+\frac{\Delta x}{2}\right)\frac{\sinh\frac{\Delta x}{2}}{\frac{\Delta x}{2}}\\ & =\cosh x \end{align*}

(2)-1

\begin{align*} \left(\cosh x\right)' & =\lim_{\Delta x\rightarrow0}\frac{\cosh(x+\Delta x)-\cosh x}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\frac{\cosh x\cosh\Delta x+\sinh x\sinh\Delta x-\cosh x}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\left(\frac{\sinh x\sinh\Delta x}{\Delta x}+\frac{\cosh x\left(\cosh\Delta x-1\right)}{\Delta x}\right)\\ & =\lim_{\Delta x\rightarrow0}\left(\frac{\sinh x\sinh\Delta x}{\Delta x}+\frac{\cosh x\sinh^{2}\Delta x}{\Delta x\left(\cos\Delta x+1\right)}\right)\\ & =-\lim_{\Delta x\rightarrow0}\frac{\sinh\Delta x}{\Delta x}\left(\sinh x+\frac{\cosh x\sinh\Delta x}{\cosh\Delta x+1}\right)\\ & =\sinh x \end{align*}

(2)-2

\begin{align*} \left(\cosh x\right)' & =\lim_{\Delta x\rightarrow0}\frac{\cosh(x+\Delta x)-\cosh x}{\Delta x}\\ & =\lim_{\Delta x\rightarrow0}\frac{1}{\Delta x}\left(2\sinh\frac{2x+\Delta x}{2}\sinh\frac{\Delta x}{2}\right)\\ & =\lim_{\Delta x\rightarrow0}\sinh\left(x+\frac{\Delta x}{2}\right)\frac{\sinh\frac{\Delta x}{2}}{\frac{\Delta x}{2}}\\ & =\sinh x \end{align*}

(3)

\begin{align*} \left(\tanh x\right)' & =\left(\frac{\sinh x}{\cosh x}\right)'\\ & =\frac{\left(\sinh x\right)'\cosh x-\sinh x\left(\cosh x\right)'}{\cosh^{2}x}\\ & =\frac{\cosh^{2}x-\sinh^{2}x}{\cosh^{2}x}\\ & =\frac{1}{\cosh^{2}x} \end{align*}

(3)-2

\begin{align*} \left(\tanh x\right)' & =\lim_{\Delta x\rightarrow0}\frac{1}{\Delta x}\left(\frac{\sinh\left(x+\Delta x\right)}{\cosh(x+\Delta x)}-\frac{\sinh x}{\cosh x}\right)\\ & =\lim_{\Delta x\rightarrow0}\frac{1}{\cosh x\cosh(x+\Delta x)\Delta x}\left(\sinh\left(x+\Delta x\right)\cosh x-\cosh(x+\Delta x)\sinh x\right)\\ & =\lim_{\Delta x\rightarrow0}\frac{\sinh\Delta x}{\cosh x\cosh(x+\Delta x)\Delta x}\\ & =\frac{1}{\cosh^{2}x} \end{align*}