Question

How do you differentiate `f(x) = x^3/(xcotx+1)` using the quotient rule?

Answer 1

`frac{d}{dx}(frac{x^3}{xcot (x)+1})=frac{2x^3sin ^2(x)cot (x)+3x^2sin ^2(x)+x^4}{sin ^2(x)(xcot (x)+1)^2}`

Explanation:

`frac{d}{dx}(frac{x^3}{xcot (x)+1})`

Applying quotient rule,

`(frac{f}{g})^'=frac{f^'cdot g-g^'cdot f}{g^2}`

`=frac{frac{d}{dx}(x^3)(xcot (x)+1)-frac{d}{dx}(xcot (x)+1)x^3}{(xcot (x)+1)^2}`

We know,
`frac{d}{dx}(x^3)=3x^2`
`frac{d}{dx}cot (x)+1=-frac{x}{sin ^2(x)}+cot(x)`

so,`=frac{3x^2(xcot (x)+1)-(-frac{x}{sin ^2(x)}+cot (x))x^3}{(xcot (x)+1)^2}`

Simplifying it,
`frac{2x^3sin ^2(x)cot (x)+3x^2sin ^2(x)+x^4}{sin ^2(x)(xcot (x)+1)^2}`