Question

Find the differential of `x^(cot x) + (2x^2 - 3)/(2x^2 - x + 2)` with respect to x.

Answer 1

`x^(cot x) + (2x^2 - 3)/(2x^2 - x + 2)`

Split the expression into two parts: u and v.

Let y = u + v,

u = x^{cot x}

v = `(2x^2 - 3)/(2x^2 - x + 2)`

Then,

`(dy)/(dx) = (du)/(dx) + (du)/(dx)`

Differentiate u = x^{cot x}

We use logarithmic differentiation:

log u = log(x^{cot x})

log u = cot x . log x

Differentiating both sides with respect to x:

`1/u (du)/(dx) = d/dx (cot x) . log x + cot x . d/dx (log x)`

`1/u (du)/(dx) = (-csc^2 x) log x + cot x (1/x)`

`(du)/(dx) = u[cotx/x - csc^2 x log x]`

`(du)/(dx) = x^cot x [cot x/x - csc^2 x log x]`

Differentiate v = `(2x^2 - 3)/(2x^2 - x + 2)`

We use quotient rule: `d/dx (f/g) = (f'g - fg')/g^2`

Let f = 2x² − 3 ⇒ f′ = 4x

Let g = 2x² − x + 2 ⇒ g′ = 4x − 1

`(dv)/(dx) = ((4x)(2x^2 - x + 2) - (2x^2 - 3)(4x - 1))/(2x^2 - x + 2)^2`

`(dv)/(dx) = ((8x^3 - 4x^2 + 8x) - (8x^3 - 2x^2 - 12x + 3))/(2x^2 - x + 2)^2`

`(dv)/(dx) = (8x^3 - 4x^2 + 8x - 8x^3 + 2x^2 + 12x - 3)/(2x^2 - x + 2)^2`

`(dv)/(dx) = (-2x^2 + 20x - 3)/(2x^2 - x + 2)^2`

Combining the two results:

`(dy)/(dx) = x^cot x [cot x/x - csc^2 x log x] + (-2x^2 + 20x - 3)/(2x^2 - x + 2)^2`