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Question
Differentiate the following w.r.t. x : `cos^-1((3cos3x - 4sin3x)/5)`
Solution
Let y = `cos^-1((3cos3x - 4sin3x)/5)`
= `cos^-1[(cos3x)(3/5) - (sin3x)(4/5)]`
Since,`(3/5)^2 + (4/5)^2`
= `(9)/(25) + (16)/(25)` = 1
we can write, `(3)/(5) = cos ∞ and (4)/(5) = sin ∞`.
∴ y = cos–1(cos3x cos∞ – sin3x sin∞)
= cos–1[cos(3x + ∞)
= 3x + ∞, where ∞ is a constant
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"(3x + ∞)`
= `3"d"/"dx"(x) + "d"/"dx"(∞)`
= 3 x 1 + 0
= 3.
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