Question

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): x / sin^n x​

Answer 1

Answer:

refer to the attachment

Answer 2

Given :-

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$\tt\longrightarrow{\dfrac{x}{sin^n x}}$

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To find :-

• Derivative of the given function.

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Solution :-

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$\: \: \: \: \: \: \: \: \bullet\bf\: \: \: {Let\: f(x) = \dfrac{x}{sin^n x}}$

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By differentiating and using quotient rule, we get

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$\implies\sf{f^n (x) = \dfrac{sin^n x \dfrac{d}{dx}x - x\dfrac{d}{dx}sin^n x}{(sin^n x)^2}}$

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$\implies\sf{f^n (x) = \dfrac{sin^n x \dfrac{d}{dx}x - x\dfrac{d}{dx}sin^n x}{sin^{2n} x}}$

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We know that,

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⠀⠀⠀⠀$\bigstar\sf{\: \: \: \: \dfrac{d}{dx}sin^n x = nsin^{n - 1} xcosx}$

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Further Calculation

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$\tt:\implies{f^n (x) = \dfrac{sin^n x 1 - x\: nsin^{n - 1} xcosx}{sin^{2n} x}}{}$

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$\tt:\implies{f^n (x) = sin^{n - 1} \bigg\lgroup \dfrac{x(sinx - nx) cosx}{sin^{2n} x} \bigg\rgroup}$

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$\tt:\implies{f^n (x) = \dfrac{(sinx - nx cosx)}{sin{n + 1} x}}$

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That is the required answer..