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

2 Answers

See the answer below:
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Apr 13, 2017

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.