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# If y =$\dfrac{1}{{\sqrt[3]{{{\text{cosec x + cot x}}}}}}$, find $\dfrac{{{\text{dx}}}}{{{\text{dy}}}}$ .

Last updated date: 21st Jul 2024
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Hint: Look into the table of derivatives of trigonometric functions for cosec x and cot x. Convert the root into power and then differentiate it.

Given Data,

y =$\dfrac{1}{{\sqrt[3]{{{\text{cosec x + cot x}}}}}}$
Transform y such that there is no cube root in the equation, for the ease of solving
$\Rightarrow$ y = $\dfrac{1}{{{{\left( {\cos {\text{ec x + cot x}}} \right)}^{\dfrac{1}{3}}}}} = {\left( {\cos {\text{ec x + cot x}}} \right)^{ - \dfrac{1}{3}}}$
Differentiating y with respect to x
$\Rightarrow$ $\dfrac{{{\text{dy}}}}{{{\text{dx}}}} = \dfrac{{\text{d}}}{{{\text{dx}}}}{\left( {{\text{cosec x + cot x}}} \right)^{ - \dfrac{1}{3}}}$

For a function f = (x + 1)$^2$, $\dfrac{{{\text{df}}}}{{{\text{dx}}}}{\text{ becomes 2(x + 1}}{{\text{)}}^{2 - 1}}\dfrac{{\text{d}}}{{{\text{dx}}}}({\text{x + 1)}}$

Similarly here,
$\dfrac{{{\text{dy}}}}{{{\text{dx}}}}$ = $- \dfrac{1}{3}{\left( {{\text{cosec x + cot x}}} \right)^{ - \dfrac{1}{3} - 1}}\dfrac{{\text{d}}}{{{\text{dx}}}}\left( {{\text{cosec x + cot x}}} \right)$

From the table of derivatives of trigonometric functions,
$\dfrac{{\text{d}}}{{{\text{dx}}}}({\text{cosec x) = - cosec(x)cot(x)}} \\ \dfrac{{\text{d}}}{{{\text{dx}}}}(\cot {\text{x) = - cose}}{{\text{c}}^2}({\text{x)}} \\ \\$

Now,
$\dfrac{{{\text{dy}}}}{{{\text{dx}}}}$ = $- \dfrac{1}{3}{\left( {{\text{cosec x + cot x}}} \right)^{ - \dfrac{4}{3}}}\left( {{\text{ - cosec x cot x - cose}}{{\text{c}}^2}{\text{ x}}} \right)$
Take –cosec x common,
$\Rightarrow$ $\dfrac{{{\text{dy}}}}{{{\text{dx}}}}$ = $\dfrac{{{\text{cosec x}}}}{3}{\left( {{\text{cosec x + cot x}}} \right)^{ - \dfrac{4}{3}}}\left( {{\text{cot x + cosec x}}} \right)$
Adding powers of similar terms, we get -------- (${{\text{a}}^{\text{m}}} \times {{\text{a}}^{\text{n}}} = {{\text{a}}^{{\text{m + n}}}}$)
$\Rightarrow$ $\dfrac{{{\text{dy}}}}{{{\text{dx}}}}$ = $\dfrac{{{\text{cosec x}}}}{3}{\left( {{\text{cosec x + cot x}}} \right)^{ - \dfrac{1}{3}}} \\ \\$
= $\dfrac{{{\text{cosec x}}}}{{3{{\left( {{\text{cosec x + cot x}}} \right)}^{\dfrac{1}{3}}}}}$