Question

Differentiate the following:
cot³x²​

Answer 1

$\implies {\cot}^{3} ( {x}^{2} )$

We have to differentiate the above expression.

$\implies \dfrac{d({\cot}^{3} ( {x}^{2} ) )}{dx}$

we will use chain rule to differentiate.

$\implies 3{\cot}^{2} ( {x}^{2} ) \times \: \dfrac{d({\cot} {x}) }{dx} \: \times \dfrac{d({x}^{2} )}{dx}$

$\implies 3{\cot}^{2} ( {x}^{2} ) \times ( - {\csc}^{2} (x) )\: \times 2x$

$\implies - 6x \: {\cot}^{2} ( {x}^{2} ) \: {\csc}^{2} (x)$

Answer :

$- 6x \: {\cot}^{2} ( {x}^{2} ) \: {\csc}^{2} (x)$

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$\large\bf\blue{Additional-Information}$

d(sinx)/dx = cosx

d(cos x)/dx = -sin x

d(cosec x)/dx = -cot x cosec x

d(tan x)/dx = sec²x

d(sec x)/dx = secx tanx

d(cot x)/dx = - cosec² x

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Answer 2

Given ,

The function is y = cot³x²

Differentiaing y wrt to x by using chain rule , we get

$\tt \hookrightarrow \frac{dy}{dx} = \frac{ {d \{cot}^{3} {(x)}^{2} \}}{dx}$

$\tt \hookrightarrow \frac{dy}{dx} = \frac{d \{ {cot {(x)}^{2} \} }^{3} }{dx}$

$\tt \hookrightarrow \frac{dy}{dx} = 3 \{ {cot {(x)}^{2} \}}^{2} \frac{d \{cot {(x)}^{2} \}}{dx}$

$\tt \hookrightarrow \frac{dy}{dx} = - 3 {cot}^{2} {(x)}^{2} {cosec}^{2} {(x)}^{2} \frac{d \{ {x}^{2} \}}{dx}$

$\tt \hookrightarrow \frac{dy}{dx} = - 3 {cot}^{2} {(x)}^{2} {cosec}^{2} {(x)}^{2} 2x$

$\tt \hookrightarrow\frac{dy}{dx} = - 6x {cot}^{2} {(x)}^{2} {cosec}^{2} {(x)}^{2}$

Remmember :

$\tt \frac{d \{ {x}^{n} \}}{dx} = n { x}^{n - 1}$

$\tt \frac{d \{ cot(x) \}}{dx} = - {cosec}^{2} (x)$

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