Find the derivatives of the functions defined as y=csc⁡xx

Step 1 Given: y=csc⁡xx We know that ddx(uv)=vdudx−udvdxv2...(1) Step 2 So, by using equation(1) derivative of given expression will be dydx=ddx(cosec xx) =xddx(cosec x)−(cosec x)ddx(x)x2 (∵ddx(cosec x)=−cosec x cot⁡x,ddx(xn)=nxn−1) =x(−cosec x cot⁡x)−(cosec x)(1)x2 =−cosec x(x cot⁡x+1)x2 Hence, derivative of given expression is equal to −cosec x(x cot⁡x+1)x2.

Expert answers to "Find the derivatives of the functions defined as..."

jernplate8

Answered question

2021-10-20

Find the derivatives of the functions defined as

y = \frac{\csc x}{x}

Answer & Explanation

Cullen

Skilled2021-10-21Added 89 answers

Step 1
Given: $y = \frac{\csc x}{x}$
We know that
$\frac{d}{d x} (\frac{u}{v}) = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}$ ...(1)
Step 2
So, by using equation(1) derivative of given expression will be
$\frac{d y}{d x} = \frac{d}{d x} (\frac{c o s e c x}{x})$
$= \frac{x \frac{d}{d x} (c o s e c x) - (c o s e c x) \frac{d}{d x} (x)}{x^{2}}$
$(∵ \frac{d}{d x} (c o s e c x) = - c o s e c x \cot x, \frac{d}{d x} (x^{n}) = n x^{n - 1})$
$= \frac{x (- c o s e c x \cot x) - (c o s e c x) (1)}{x^{2}}$
$= \frac{- c o s e c x (x \cot x + 1)}{x^{2}}$
Hence, derivative of given expression is equal to $\frac{- c o s e c x (x \cot x + 1)}{x^{2}}$ .

Jeffrey Jordon

Expert2022-03-24Added 2605 answers

Answer is given below (on video)

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Find the derivatives of the functions defined as y=\frac{\csc x}{x}

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