Find dy/dx, when y = x^n + n^x + x^x +n^n

Question

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

y = xⁿ + n^x + x^x + nⁿ

Answer 1

let y = xⁿ + n^x + x^x + nⁿ

⇒ y = a + b + c + m

where a= xⁿ; b = n^x ; c = x^x ; m= nⁿ

  \(\Bigg\{\) 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 = xⁿ

Taking log both the sides:

⇒ log a= log xⁿ

⇒ log a= n log x

{log x^a = a log x}

⇒ log a= n log x {log e -1}

Differentiating with respect to x:

   \(\Bigg\{\) Using chain rule, \(\cfrac{d(au)}{d\text x}\) = a\(\cfrac{du}{d\text x}\) where a is any constant and u is any variable\(\Bigg\}\)

b = n^x

Taking log both the sides:

⇒ log b= log n^x

⇒ log b= x log n

{log x^a = a log x}

Differentiating with respect to x:

    \(\Bigg\{\) Using chain rule, \(\cfrac{d(au)}{d\text x}\) = a\(\cfrac{du}{d\text x}\) where a is any constant and u is any variable\(\Bigg\}\)

c = x^x

Taking log both the sides:

⇒ log c= log x^x

⇒ log c= x log x

{log x^a = a log x}

Differentiating with respect to x: