Derivative of Tan ax

What is the Derivative of Tan ax ?

The derivative of tan ax is a.(1+tan² ax).

$(tanax{)}^{\prime }=a.(1+ta{n}^{2}ax)$

$\frac{d}{dx}(tanax)=a.(1+ta{n}^{2}ax)$

Proof of the Derivative of Tan ax

Way 1

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{tana(x+h)-tanax}{h}$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{tan(ax+ah)-tanax}{h}$

$\boxed{tan(p+q)=\frac{tanp+tanq}{1-tanp.tanq}}$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\frac{tanax+tanah}{1-tanax.tanah}-tanax}{h}$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\frac{tanax+tanah-tanax.(1-tanax.tanah)}{1-tanax.tanah}}{h}$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\frac{\cancel{tanax}+tanah-\cancel{tanax}+ta{n}^{2}ax.tanah}{1-tanax.tanah}}{h}$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\frac{tanah+ta{n}^{2}ax.tanah}{1-tanax.tanah}}{h}$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\frac{tanah.(1+ta{n}^{2}ax)}{1-tanax.tanah}}{h}$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }[\frac{1}{h}.\frac{tanah.(1+ta{n}^{2}ax)}{1-tanax.tanah}]$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{tanah.(1+ta{n}^{2}ax)}{h.(1-tanax.tanah)}$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{a.tanah.(1+ta{n}^{2}ax)}{a.h.(1-tanax.tanah)}$

$(tanax{)}^{\prime }=a.\underset{h\to 0}{\lim }\frac{tanah.(1+ta{n}^{2}ax)}{ah.(1-tanax.tanah)}$

$(tanax{)}^{\prime }=a.\underset{h\to 0}{\lim }(\frac{tanah}{ah}.\frac{1+ta{n}^{2}ax}{1-tanax.tanah})$

$(tanax{)}^{\prime }=a.\underset{h\to 0}{\lim }\frac{tanah}{ah}.\underset{h\to 0}{\lim }\frac{1+ta{n}^{2}ax}{1-tanax.tanah}$

$h\to 0(ah=h)$

$(tanax{)}^{\prime }=a.\underset{h\to 0}{\lim }\frac{tanh}{h}.\underset{h\to 0}{\lim }\frac{1+ta{n}^{2}ax}{1-tanax.tanh}$

$\boxed{\underset{t\to 0}{\lim }\frac{tant}{t}=1}$

$(tanax{)}^{\prime }=a.1.\frac{1+ta{n}^{2}ax}{1-tanax.tan0}$

$\boxed{tan0=0}$

$(tanax{)}^{\prime }=a.1.\frac{1+ta{n}^{2}ax}{1-tanax.0}$

$(tanax{)}^{\prime }=a.1.\frac{1+ta{n}^{2}ax}{1-0}$

$(tanax{)}^{\prime }=a.1.\frac{1+ta{n}^{2}ax}{1}$

$(tanax{)}^{\prime }=a.1.(1+ta{n}^{2}ax)$

$(tanax{)}^{\prime }=a.(1+ta{n}^{2}ax)$

Way 2

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{tana(x+h)-tanax}{h}$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{tan(ax+ah)-tanax}{h}$

$\boxed{tanx=\frac{sinx}{cosx}}$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\frac{sin(ax+ah)}{cos(ax+ah)}-\frac{sinax}{cosax}}{h}$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\frac{sin(ax+ah).cosax-cos(ax+ah).sinax}{cosax.cos(ax+ah)}}{h}$

$\boxed{sinp.cosq-cosp.sinq=sin(p-q)}$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\frac{sin(\cancel{ax}+ah-\cancel{ax})}{cosax.cos(ax+ah)}}{h}$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{\frac{sinah}{cosax.cos(ax+ah)}}{h}$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }[\frac{1}{h}.\frac{sinah}{cosax.cos(ax+ah)}]$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{sinah}{h.cosax.cos(ax+ah)}$

$(tanax{)}^{\prime }=\underset{h\to 0}{\lim }\frac{a.sinah}{a.h.cosax.cos(ax+ah)}$

$(tanax{)}^{\prime }=a.\underset{h\to 0}{\lim }\frac{sinah}{ah.cosax.cos(ax+ah)}$

$(tanax{)}^{\prime }=a.\underset{h\to 0}{\lim }[\frac{sinah}{ah}.\frac{1}{cosax.cos(ax+ah)}]$

$(tanax{)}^{\prime }=a.\underset{h\to 0}{\lim }\frac{sinah}{ah}.\underset{h\to 0}{\lim }\frac{1}{cosax.cos(ax+ah)}$

$h\to 0(ah=h)$

$(tanax{)}^{\prime }=a.\underset{h\to 0}{\lim }\frac{sinh}{h}.\underset{h\to 0}{\lim }\frac{1}{cosax.cos(ax+h)}$

$\boxed{\underset{t\to 0}{\lim }\frac{sint}{t}=1}$

$(tanax{)}^{\prime }=a.1.\frac{1}{cosax.cos(ax+0)}$

$(tanax{)}^{\prime }=a.1.\frac{1}{cosax.cosax}$

$(tanax{)}^{\prime }=\frac{a.1.1}{co{s}^{2}ax}$

$(tanax{)}^{\prime }=\frac{a}{co{s}^{2}ax}$

$(tanax{)}^{\prime }=a.\frac{1}{co{s}^{2}ax}$

$(tanax{)}^{\prime }=a.(\frac{1}{cosax}{)}^2$

$\boxed{\frac{1}{cosx}=secx}$

$(tanax{)}^{\prime }=a.se{c}^{2}ax$

$(tanax{)}^{\prime }=a.(1+ta{n}^{2}ax)=\frac{a}{co{s}^{2}ax}=a.se{c}^{2}ax$