Question

(i) tan² x

(ii) tan (2x + 1)

(iii) tan 2x

(iv)

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\frac{d}{dx}\left(f\left(x\right)\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h} \\ =\underset{h\to 0}{\lim }\frac{{\tan }^2\left(x+h\right)-{\tan }^{2}x}{h} \\ =\underset{h\to 0}{\lim }\frac{\left[\tan \left(x+h\right)+\tan x\right]\left[\tan \left(x+h\right)-\tan x\right]}{h} \\ =\underset{h\to 0}{\lim }\frac{\left[{\displaystyle \frac{\sin \left(x+h\right)}{\cos \left(x+h\right)}}+{\displaystyle \frac{\sin x}{\cos x}}\right]\left[{\displaystyle \frac{\sin (x+h)}{\cos (x+h)}-\frac{\sin x}{\cos x}}\right]}{h} \\ =\underset{h\to 0}{\lim }\frac{\left[\sin \left(x+h\right)\cos x+\cos \left(x+h\right)\sin x\right]\left[\sin \left(x+h\right)\cos x-\cos \left(x+h\right)\sin x\right]}{h{\cos }^{2}x{\cos }^2\left(x+h\right)} \\ =\underset{h\to 0}{\lim }\frac{\left[\sin \left(2x+h\right)\right]\left[\sin h\right]}{h{\cos }^{2}x{\cos }^2\left(x+h\right)} \\ =\frac{1}{{\cos }^{2}x}\underset{h\to 0}{\lim }\sin \left(2x+h\right)\underset{h\to 0}{\lim }\frac{\sin h}{h}\underset{h\to 0}{\lim }\frac{1}{{\cos }^2\left(x+h\right)} \\ =\frac{1}{{\cos }^{2}x}\sin \left(2x\right)\left(1\right)\frac{1}{{\cos }^{2}x} \\ =\frac{1}{{\cos }^{2}x}2\sin x\cos x\frac{1}{{\cos }^{2}x} \\ =2\times \frac{\sin x}{\cos x}\times \frac{1}{{\cos }^{2}x} \\ =2\tan x{\sec }^{2}x \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)\frac{d}{dx}\left(f\left(x\right)\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h} \\ =\underset{h\to 0}{\lim }\frac{\tan \left(2x+2h+1\right)-\tan \left(2x+1\right)}{h} \\ =\underset{h\to 0}{\lim }\frac{{\displaystyle \frac{\mathrm{si}n\left(2x+2h+1\right)}{\cos \left(2x+2h+1\right)}}-{\displaystyle \frac{\sin \left(2x+1\right)}{\cos \left(2x+1\right)}}}{h} \\ =\underset{h\to 0}{\lim }\frac{\mathrm{si}n\left(2x+2h+1\right)\cos \left(2x+1\right)-\cos \left(2x+2h+1\right)\sin \left(2x+1\right)}{h\cos \left(2x+2h+1\right)\cos \left(2x+1\right)} \\ =\underset{h\to 0}{\lim }\frac{\sin \left(2x+2h+1-2x-1\right)}{h\cos \left(2x+2h+1\right)\cos \left(2x+1\right)} \\ =\frac{1}{\cos \left(2x+1\right)}\underset{h\to 0}{\lim }\frac{\sin \left(2h\right)}{2h}\times 2\underset{h\to 0}{\lim }\frac{1}{\cos \left(2x+2h+1\right)} \\ =\frac{1}{\cos \left(2x+1\right)}\times 2\times \frac{1}{\cos \left(2x+1\right)} \\ =\frac{2}{{\cos }^2\left(2x+1\right)} \\ =2{\sec }^2\left(2x+1\right) \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iii}\right)\frac{d}{dx}\left(f\left(x\right)\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h} \\ =\underset{h\to 0}{\lim }\frac{\tan \left(2x+2h\right)-\tan \left(2x\right)}{h} \\ =\underset{h\to 0}{\lim }\frac{{\displaystyle \frac{\mathrm{si}n\left(2x+2h\right)}{\cos \left(2x+2h\right)}}-{\displaystyle \frac{\sin \left(2x\right)}{\cos \left(2x\right)}}}{h} \\ =\underset{h\to 0}{\lim }\frac{\mathrm{si}n\left(2x+2h\right)\cos \left(2x\right)-\cos \left(2x+2h\right)\sin \left(2x\right)}{h\cos \left(2x+2h\right)\cos \left(2x\right)} \\ =\underset{h\to 0}{\lim }\frac{\sin \left(2x+2h-2x\right)}{h\cos \left(2x+2h\right)\cos \left(2x\right)} \\ =\frac{1}{\cos 2x}\underset{h\to 0}{\lim }\frac{\sin \left(2h\right)}{2h}\times 2\times \underset{h\to 0}{\lim }\frac{1}{\cos \left(2x+2h\right)} \\ =\frac{1}{\cos 2x}\times 2\times \frac{1}{\cos 2x} \\ =\frac{2}{{\cos }^2\left(2x\right)} \\ =2{\sec }^2\left(2x\right) \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iv}\right)\frac{d}{dx}\left(f\left(x\right)\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h} \\ =\underset{h\to 0}{\lim }\frac{\sqrt{\tan \left(x+h\right)}-\sqrt{\tan x}}{h}\times \frac{\sqrt{\tan \left(x+h\right)}+\sqrt{\tan x}}{\sqrt{\tan \left(x+h\right)}+\sqrt{\tan x}} \\ =\underset{h\to 0}{\lim }\frac{\tan \left(x+h\right)-\tan x}{h\left(\sqrt{\tan \left(x+h\right)}+\sqrt{\tan x}\right)} \\ =\underset{h\to 0}{\lim }\frac{{\displaystyle \frac{\sin \left(x+h\right)}{\cos \left(x+h\right)}}-{\displaystyle \frac{\sin x}{\cos x}}}{h\left(\sqrt{\tan \left(x+h\right)}+\sqrt{\tan x}\right)} \\ =\underset{h\to 0}{\lim }\frac{\sin \left(x+h\right)\cos x-\cos (x+h)\sin x}{h\left(\sqrt{\tan \left(x+h\right)}+\sqrt{\tan x}\right)\cos \left(x+h\right)\cos x} \\ =\underset{h\to 0}{\lim }\frac{\sin h}{h\left(\sqrt{\tan \left(x+h\right)}+\sqrt{\tan x}\right)\cos \left(x+h\right)\cos x} \\ =\underset{h\to 0}{\lim }\frac{\sin h}{h}\underset{h\to 0}{\lim }\frac{1}{\left(\sqrt{\tan \left(x+h\right)}+\sqrt{\tan x}\right)\cos \left(x+h\right)\cos x} \\ =\left(1\right)\frac{1}{2\sqrt{\tan x}{\cos }^{2}x} \\ =\frac{{\sec }^{2}x}{2\sqrt{\tan x}} \end{gathered}$$