Question

Differentiate the following w.r.t. x : `(x^5.tan^3 4x)/(sin^2 3x)`

Answer 1

Let y = `(x^5.tan^3 4x)/(sin^2 3x)`

Then log y = `log[(x^5.tan^3 4x)/(sin^23x)]`

= logx⁵ + log tan³4x – log sin²3x

= 5logx + 3log (tan4x) – 2log (sin3x)

Differentiating both sides w.r.t. x, we get

`(1)/y."dy"/"dx" = 5"d"/"dx"(logx) + 3"d"/"dx"[log(tan4x)] - 2"d"/"dx"[log(sin3x)]`

= `5 xx (1)/x + 3 xx (1)/(tan4x)."d"/"dx"(tan4x) - 2 xx (1)/(sin3x)."d"/"dx"(sin3x)`

= `5/x + 3 xx (1)/(tan4x) xx sec^2  4x."d"/"dx"(4x) - 2 xx (1)/(sin3x) xx cos3x."d"/"dx"(3x)`

= `5/x + 3.(cos4x)/(sin4x) xx (1)/(cos^2 4x) xx 4 - 2cot3x xx 3`

= `5/x + (24)/(2sin4x.cos4x) - 6cot3x`

∴ `"dy"/"dx" = y[5/x + 24/(sin8x) - 6cot3x]`

= `(x^5.tan^3 4x)/(sin^2 3x)[5/x + 24"cosec"8x - 6cot3x]`.