Derivative of cot^2x by First Principle

The derivative of cot^2x is -2cotx cosec²x. Here we find the derivative of cot square x by the first principle of derivatives.

Differentiation of cot square x by the first principle will be computed as follows.

Derivative of cot square x by First Principle

By first principle, the derivative of f(x) is given by the limit below.

$\dfrac{d}{dx}(f(x))$ = lim_h→0 $\dfrac{f(x+h)-f(x)}{h}$ …(I)

Put f(x)=cot²x. Therefore,

$\dfrac{d}{dx}(\cot^2 x)$ = lim_h→0 $\dfrac{\cot^2(x+h)-\cot^2x}{h}$

⇒ $\dfrac{d}{dx}(\cot^2 x)$ = lim_h→0 $\Big\{ [\cot(x+h)+\tan x] \times$ $\dfrac{\cot(x+h)-\cot x}{h} \Big\}$ by the algebraic identity a²-b² = (a+b)(a-b).

⇒ $\dfrac{d}{dx}(\cot^2 x)$ = lim_h→0 $[\cot(x+h)+ \cot x]$ × lim_h→0 $\dfrac{\cot(x+h)-\cot x}{h}$ using the product rule of limits.

Note that if we put f(x)=cot x in the above formula (I), then the limit $\lim\limits_{h \to 0}\frac{\cot(x+h)-\cot x}{h}$ will be equal to the derivative of cot x, and we know that d/dx(cot x) = -cosec²x. Thus from above we obtain that

∴ $\dfrac{d}{dx}(\cot^2 x)$ =2 cotx × (-cosec²x) = -2cotx cosec²x.

Therefore, the derivative of cot^2x by the first principle is -2cot x cosec²x.

More Derivatives:

Derivative of tan2x	Derivative of $\sqrt{\sin x}$
Derivative of 1/x2	Derivative of tan2x
Derivative of cos3x	Derivative of 1/x

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