Find dy/dx, when y = (tan x)^log x + cos^2(π/4)

Question

Find \(\cfrac{dy}{d\text x}\), when

y = (tan x)^{log x} + cos²\(\big(\cfrac{\pi}4\big)\)

Answer 1

Let y = (tan x)^{log x} + cos²\(\big(\cfrac{\pi}4\big)\)

⇒ y = a + b

where, a = (tan x)^{log x}; b = cos²\(\big(\cfrac{\pi}4\big)\)

  \(\Bigg\{\) Using chain rule, \(\cfrac{d(u +a)}{d\text x}=\cfrac{du}{d\text x}+\cfrac{da}{d\text x}\) where a and u are any variables \(\Bigg\}\)

a= (tan x)^{log x}

Taking log both the sides:

⇒ log a= log (tan x)^{log x}

⇒ log a= log x . log (tan x)

{log x^a = a log x}

Differentiating with respect to x:

b = cos²\(\big(\cfrac{\pi}4\big)\) 

Differentiating with respect to x: