Find $\dfrac{dy}{dx}$, if $(\cos x)^{y}=(\cos y)^{x}$.

$\dfrac{y\tan x+\ln(\cos y)}{x\tan y+\ln(\cos x)}$

CBSE Class 12 · 2024 · DifferentiationShort Answer

Find

\frac{d y}{d x}

, if

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

Answer: $\frac{y tan x + ln ( cos y )}{x tan y + ln ( cos x )}$

Take natural logarithm of both sides:

y ln (cos x) = x ln (cos y)

Differentiate both sides w.r.t.

x

(product rule on each side):

\frac{d y}{d x} ln (cos x) + y \cdot \frac{- sin x}{cos x} = ln (cos y) + x \cdot \frac{- sin y}{cos y} \frac{d y}{d x}

\frac{d y}{d x} ln (cos x) - y tan x = ln (cos y) - x tan y \frac{d y}{d x}

Collect the

\frac{d y}{d x}

terms:

\frac{d y}{d x} [ln (cos x) + x tan y] = ln (cos y) + y tan x

\frac{d y}{d x} = \frac{y tan x + ln ( cos y )}{x tan y + ln ( cos x )}

CBSE Class 12 Board · 2024 · Subjective