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