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Question
Differentiate the following w.r.t. x :
`cos^-1[(3cos(e^x) + 2sin(e^x))/sqrt(13)]`
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Solution
y = `cos^-1((3cos(e^x) + 2sin(e^x))/sqrt(13))`
y = `cos^-1(cos(e^x).(3)/sqrt(13) + sin(e^x)2/sqrt(13))`
Put, `(3)/sqrt(13) = cos α, (2)/sqrt(13) = sin α`
Also,
sin2α + cos2α = `(9)/(13) + (4)/(13)` = 1
And,
tan α = `(2)/(3) ∴ α = "tan"^-1(2/3)`
y = cos–1(cos ex. cos α + sin ex. sin α)
y = cos–1[cos(ex – α)] ...[∵ cos–1x.(cos x) = x]
y = ex – α
y = `e^x – tan^-1(2/3)`
Differentiating w.r.t. x, we get
`dy/dx = d/dx [e^x - tan^-1(2/3)]`
`dy/dx` = ex – 0
`dy/dx` = ex
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