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Question
Differentiate the function with respect to x:
`(5x)^(3cos 2x)`
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Solution
Let y = `(5x)^(3cos 2x)`
Taking log on both sides, we get
log y = 3 cos 2x log (5x) = 3 cos 2x [log 5 + log x]
log y = 3 cos 2x log 5 + 3 cos 2x log x ....(1)
Differentiating (1) with respect to x, we get
`1/y dy/dx = 3 log 5 (-sin 2x)* 2 + (3 cos 2x)/x + 3 log x (-2 sin 2x)`
= `-6 log 5 sin 2x + (3 cos 2x)/x - 6 log x sin 2x`
`dy/dx = (5x)^(3cos 2x) [(3 cos 2x)/x - 6 (log 5 + log x) sin 2x]`
= `(5x)^(3 cos 2x) [(3 cos 2x)/x - 6 log 5x sin 2x]`
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