Question

How do you differentiate `y = ((sin(x))^6 (tan(x))^2) / (x^2 + 2)^2`?

Answer 1

See the answer below:

Answer 2

By Log Differentiation,

`y'=(8 cot x+2tan x-(4x)/(x^2+2))(sin^6x cdot tan^2 x)/(x^2+2)^2`

Explanation:

You can also use Logarithmic Differentiation.

By `tan x=sin x/cos x`,

#y=(sin^6x cdot tan^2 x)/(x^2+2)^2 =(sin^6x cdot (sin^2 x)/(cos^2 x))/(x^2+2)^2 =(sin^8 x)/(cos^2x(x^2+2)^2)#

By taking the natural log of both sides,

`Rightarrow ln y=ln((sin^8 x)/(cos^2 x(x^2+2)^2))`

By Log Properties: `ln(x cdot y)=ln x + ln y` and `ln(x/y)=ln x - ln y,`

`Rightarrow ln y=ln(sin^8 x)-ln(cos^2 x)-ln(x^2+2)^2`

By Log Property: `ln x^r=r ln x`,

`Rightarrow ln y=8ln(sin x)-2ln(cos x)-2ln(x^2+2)`

By differentiating with respect to `x` using `[ln(g(x))]'=(g'(x))/g(x)`,

`Rightarrow (y')/y=8 cos x/sin x-2(-sin x)/(cos x)-2 (2x)/(x^2+2)`

By cleaning up a bit,

`Rightarrow (y')/y=8 cot x+2tan x-(4x)/(x^2+2)`

By multiplying both sides by `y`,

`Rightarrow y'=(8 cot x+2tan x-(4x)/(x^2+2))(sin^6x cdot tan^2 x)/(x^2+2)^2`

I hope that this was clear.