Finding the derivative of $ g(x) = tan(3x) $ using the definition

derivativeslimits

I was asked to find the derivative of $tan(3x)$ using the limit definition

I am bit stuck at the steps, can anyone please explain ?

Thank you so much , would be a great help !

Best Answer

hint

$$\tan(3(x+h))-\tan(3x)=(1+\tan(3x+3h)\tan(3x))\tan(3h)$$

and $$\lim_{h\to 0}\frac{\tan(3h)}{h}=3$$

Related Solutions

Calculus – Finding the Derivative of |x| Using the Limit Definition

$\dfrac{d}{dx}|x|$

$=\lim\limits_{\Delta x\to 0}\dfrac{|x+\Delta x|-|x|}{\Delta x}$

$=\lim\limits_{\Delta x\to 0}\dfrac{(|x+\Delta x|-|x|)(|x+\Delta x|+|x|)}{\Delta x(|x+\Delta x|+|x|)}$

$=\lim\limits_{\Delta x\to 0}\dfrac{|x+\Delta x|^2-|x|^2}{\Delta x(|x+\Delta x|+|x|)}$

$=\lim\limits_{\Delta x\to 0}\dfrac{(x+\Delta x)^2-x^2}{\Delta x(|x+\Delta x|+|x|)}$

$=\lim\limits_{\Delta x\to 0}\dfrac{x^2+2x\Delta x+(\Delta x)^2-x^2}{\Delta x(|x+\Delta x|+|x|)}$

$=\lim\limits_{\Delta x\to 0}\dfrac{2x+\Delta x}{|x+\Delta x|+|x|}$

$=\dfrac{x}{|x|}$

$=\dfrac{x|x|}{|x|^2}$

$=\dfrac{x|x|}{x^2}$

$=\dfrac{|x|}{x}$

[Math] Using the definition of derivative to find $\tan^2x$

Using first principle, the derivative of any function $f(x)$ is given as $$\frac{d(f(x))}{dx}=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$ Hence, derivative of $\tan^2 x$ is given as $$\frac{d(\tan^2 x)}{dx}=\lim_{h\to 0}\frac{\tan^2(x+h)-\tan^2(x)}{h}$$ $$=\lim_{h\to 0}\frac{(\tan(x+h)-\tan(x))(\tan(x+h)+\tan(x))}{h}$$$$=\lim_{h\to 0}\frac{\tan(x+h)-\tan(x)}{h}\times \lim_{h\to 0}(\tan(x+h)+\tan(x))$$ $$=\lim_{h\to 0}\frac{\frac{\sin(x+h)}{\cos(x+h)}-\frac{\sin(x)}{\cos(x)}}{h}\times \lim_{h\to 0}(\tan(x+h)+\tan(x))$$ $$=\lim_{h\to 0}\frac{\sin(x+h)\cos x-\cos(x+h)\sin x}{h\cos(x+h)\cos x}\times (\tan x+\tan x)$$ $$=2\tan x\lim_{h\to 0}\frac{\sin(x+h-x)}{h}\times \lim_{h\to 0}\frac{1}{\cos(x+h)\cos x}$$ $$=2\tan x\lim_{h\to 0}\frac{\sin h}{h}\times \frac{1}{\cos^2 x}$$ $$=2\tan x \times 1\times \sec^2 x$$ $$=\color{blue}{2\tan x \sec^2x}$$

Best Answer

Related Solutions

Calculus – Finding the Derivative of |x| Using the Limit Definition

[Math] Using the definition of derivative to find $\tan^2x$

Related Question