Question

Find the derivation of $\sqrt{tanx}$ with respect to x using first principle.

Answer 1

$a\to f^{\prime }\left(a\right)=\underset{h\to 0}{lim}\frac{f(a+h)-f\left(a\right)}{h}=\underset{h\to 0}{lim}\frac{\sqrt{\tan (a+h)}-\sqrt{\tan a}}{h}\times \frac{\sqrt{\tan (a+h)}+\sqrt{\tan a}}{\sqrt{\tan (a+h)}+\sqrt{\tan a}}$

$=\underset{h\to 0}{lim}\frac{\tan (a+h)-\tan a}{h\left(2\sqrt{\tan a}\right)}=\underset{h\to 0}{lim}\left(\frac{\tan a+\tan h-\tan a}{1-\tan a\tan h}\right)\frac{1}{2h\sqrt{\tan a}}$

$=\underset{h\to 0}{lim}\frac{\tan h+{\tan }^{2}a\tan h}{2h\sqrt{\tan a}}=\underset{h\to 0}{lim}\frac{\tan h(1+{\tan }^{2}a)}{42\sqrt{\tan a}}=\frac{{\sec }^{2}a}{2\sqrt{\tan a}}$