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Differential Equations

Question

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Differentiate the following function with respect to x:
$left parenthesis log space straight x right parenthesis to the power of straight x plus straight x to the power of log straight x$

Solution

$Let space straight y space equals left parenthesis logx right parenthesis to the power of straight x plus straight x to the power of logx space space space space space space space space... left parenthesis 1 right parenthesis Now space let space straight y subscript 1 equals left parenthesis log space straight x right parenthesis to the power of straight x space and space straight y subscript 2 equals straight x to the power of logx rightwards double arrow space space straight y space equals space straight y subscript 1 plus straight y subscript 2 space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis$
Differentiating (2), w.r.t.x,
$dy over dx equals dy subscript 1 over dx plus dy subscript 2 over dx space space space space space space space space space space... left parenthesis 3 right parenthesis$

Now consider

straight y subscript 1 space equals space left parenthesis log space straight x right parenthesis to the power of straight x

Taking log on both sides,

logy subscript 1 space equals space straight x space log left parenthesis log space straight x right parenthesis

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

1 over straight y subscript 1 dy subscript 1 over dx equals straight x cross times 1 over logx cross times 1 over straight x plus 1 cross times log left parenthesis log space straight x right parenthesis rightwards double arrow space space dy subscript 1 over dx space equals space straight y subscript 1 open parentheses 1 over logx plus log left parenthesis log space straight x right parenthesis close parentheses rightwards double arrow dy subscript 1 over dx equals left parenthesis logx right parenthesis to the power of straight x open parentheses fraction numerator 1 over denominator log space straight x end fraction plus log space left parenthesis log space straight x right parenthesis close parentheses space space space space space space... left parenthesis 4 right parenthesis

Now comma space consider space straight y subscript 2 space equals straight x to the power of logx logy subscript 2 space equals space left parenthesis log space straight x right parenthesis left parenthesis log space straight x right parenthesis space equals left parenthesis log space straight x right parenthesis squared Differentiating space straight w. straight r. straight t. straight x comma space we space get 1 over straight y subscript 2 dy subscript 2 over dx equals 2 left parenthesis log space straight x right parenthesis cross times 1 over straight x rightwards double arrow space dy subscript 2 over dx space equals straight y subscript 2 open parentheses fraction numerator 2 logx over denominator straight x end fraction close parentheses equals straight x to the power of logx open parentheses fraction numerator 2 logx over denominator straight x end fraction close parentheses... left parenthesis 5 right parenthesis

Using equations (3), (4) and (5), we get:

dy over dx equals left parenthesis log space straight x right parenthesis to the power of straight x open square brackets 1 over logx plus log left parenthesis log space straight x right parenthesis close square brackets plus straight x to the power of logx open parentheses fraction numerator 2 logx over denominator straight x end fraction close parentheses

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