Find the derivatives of the following:xy = yx

Find the derivatives of the following:
x^y = y^x

Sum

x^y = y^x

Taking log on both sides

log x^y = log y^x

y log x = x log y

Differentiate with respect to x

`y xx 1/x + (log x) ("d"y)/("d"x) = x * 1/y xx ("d"y)/("d"x) + (log y) 1`

`y/x + (log x) ("d"y)/("d"x) = x/y * ("d"y)/("d"x) + log y`

`(log ) ("d"y)/("d"x) - x/y ("d"y)/("d"x) = log y - y/x`

`(log x - x/y) ("d"y)/("d"x) = log y - y/x`

`(y log x - x)/y * ("d"y)/("d"x) = (x log y - y)/x`

`("d"y)/("d"x) = y/x * (x log y - y)/(y log x - x)`

`("d"y)/("d"x) = (y (x log y - y))/(x(y log x - x))`

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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 4 | Page 176

Find the derivatives of the following functions with respect to corresponding independent variables:

g(t) = 4 sec t + tan t

Find the derivatives of the following functions with respect to corresponding independent variables:

y = `x/(sin x + cosx)`

Find the derivatives of the following functions with respect to corresponding independent variables:

y = x sin x cos x

Find the derivatives of the following functions with respect to corresponding independent variables:

y = log₁₀ x

Differentiate the following:
y = cos (a³ + x³)

Differentiate the following:
y = e^–mx

Differentiate the following:
y = (2x – 5)⁴ (8x² – 5)^–3

Differentiate the following:

y = `(x^2 + 1) root(3)(x^2 + 2)`

Differentiate the following:
y = tan (cos x)

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

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

Find the derivatives of the following:

(cos x)^{log x}

Find the derivatives of the following:

x = a (cos t + t sin t); y = a (sin t – t cos t)

Find the derivatives of the following:

If y = `(cos^-1 x)^2`, prove that `(1 - x^2) ("d"^2y)/("d"x)^2 - x ("d"y)/("d"x) - 2` = 0. Hence find y₂ when x = 0