Find the derivatives of the following:y = xlogx+(logx)x - Mathematics

Find the derivatives of the following:
y = `x^(logx) + (logx)^x`

Sum

y = `x^(logx) + (logx)^x`

Let u = x^{log x}, v = (log x)^x

log u = log x^{log x}

log u = (log x)(log x)

log u = (log x)²

`1/u * ("d"u)/("d"x) = 2 log x xx 1/x`

`("d"u)/("d"x) = 2u (logx)/x` ........(1)

v = (log x)^x

log v = log (log x)^x

log v = x log (log x)

`1/"v" * ("dv")/("d"x) = x xx 1/logx xx 1/x + log(log x) xx 1`

`("dv")/("d"x) = "v"[1/logx + log(logx)]` ........(2)

y = u + v

`("d"y)/("d"x) = ("d"u)/("d"x) + ("dv")/("d"x)`

`("d"y)/("d"x) = 2u logx/x + "v"[1/logx + log (log x)]`

By equation (1) and (2)

`("d"y)/("d"x) = 2 x^(logx) * logx/x + (logx)^x [1/logx + log(logx)]`

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Differentiation Rules

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Chapter 10: Differential Calculus - Differentiability and Methods of Differentiation - Exercise 10.4 [Page 176]

Chapter 10 Differential Calculus - Differentiability and Methods of Differentiation
Exercise 10.4 | Q 2 | Page 176

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y = tan θ (sin θ + cos θ)

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y = e^–mx

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s(t) = `root(4)(("t"^3 + 1)/("t"^3 - 1)`

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y = `sqrt(x + sqrt(x + sqrt(x)`

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y = `sin(tan(sqrt(sinx)))`

Find the derivatives of the following:

(cos x)^{log x}