Find the derivatives of the following: (cos x)log x

Find the derivatives of the following:

(cos x)^{log x}

Sum

y = (cos x)^{log x}

Taking log on both sides

log y = log (cos x)^{log x}

log y = (log x) log (cos x)

Differentiating with respect to x

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

`1/y ("d"y)/("d"x) = - tan x (log x) + (log(cosx))/x`

`("d"y)/("d"x) = y [(log(cos x))/x - tan x (log x)]`

`("d"y)/("d"x) = (cos x)^(logx) [(log(cos x))/x - tan x (log x)]`

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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 5 | Page 176

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

y = cos x – 2 tan x

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 = `(tanx - 1)/secx`

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

y = sin x⁰

Differentiate the following:
y = `"e"^sqrt(x)`

Differentiate the following:
F(x) = (x³ + 4x)⁷

Differentiate the following:

h(t) = `("t" - 1/"t")^(3/2)`

Differentiate the following:
y = `sqrt(x + sqrt(x + sqrt(x)`

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

Find the derivatives of the following:

`tan^-1 ((cos x + sin x)/(cos x - sin x))`

Find the derivatives of the following:

If y = `(sin^-1 x)/sqrt(1 - x^2)`, show that (1 – x²)y₂ – 3xy₁ – y = 0