Differentiate the following function with respect to x : X^(sin x - cos x) + (x^2 - 1)/(x^2 + 1)

Question

Differentiate the following function with respect to x :

x^{(sin x - cos x)} + \(\cfrac{\text x^2-1}{\text x^2+1}\)

Answer 1

Let y =  x^{(sin x - cos x)} + \(\cfrac{\text x^2-1}{\text x^2+1}\)

⇒ y = a + b

where a = x^{(sin x - cos x)}; b = \(\cfrac{\text x^2-1}{\text x^2+1}\)

{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}

a = x^{(sin x - cos x)}

Taking log both the sides:

⇒ log a = log x^{(sin x - cos x)}

⇒ log a  = (sin x - cos x) log x

{log x^a = a log x}

Differentiating with respect to x:

{\(\cfrac{d(log\,u)}{d\text x}=\cfrac{1}{u}\cfrac{du}{d\text x};\) 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}

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