Question

Differentiate the following w.r.t. x : `cos^-1((3cos3x - 4sin3x)/5)`

Answer 1

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.