What is the derivative of (x^2+1^{2x-x^2}) ?

The derivative of (x^2+1^{2x-x^2}) is (ln(x^2+1)(2-2x)+(2x(2x-x^2))/(x^2+1))(x^2+1)^{2x-x^2}

What is the first derivative of (x^2+1^{2x-x^2}) ?

The first derivative of (x^2+1^{2x-x^2}) is (ln(x^2+1)(2-2x)+(2x(2x-x^2))/(x^2+1))(x^2+1)^{2x-x^2}

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

\frac{d}{d x} ((x^{2} + 1)^{2 x - x^{2}})

(ln (x^{2} + 1) (2 - 2 x) + \frac{2 x ( 2 x - x ^{2} )}{x ^{2} + 1}) (x^{2} + 1)^{2 x - x^{2}}

Solution steps

\frac{d}{d x} ((x^{2} + 1)^{2 x - x^{2}})

Apply exponent rule:

a^{b} = e^{b l n (a)}

(x^{2} + 1)^{2 x - x^{2}} = e^{(2 x - x^{2}) l n (x^{2} + 1)}

= \frac{d}{d x} (e^{(2 x - x^{2}) l n (x^{2} + 1)})

Apply the chain rule:

e^{(2 x - x^{2}) l n (x^{2} + 1)} \frac{d}{d x} ((2 x - x^{2}) ln (x^{2} + 1))

= e^{(2 x - x^{2}) l n (x^{2} + 1)} \frac{d}{d x} ((2 x - x^{2}) ln (x^{2} + 1))

\frac{d}{d x} ((2 x - x^{2}) ln (x^{2} + 1)) = ln (x^{2} + 1) (2 - 2 x) + \frac{2 x ( 2 x - x ^{2} )}{x ^{2} + 1}

= e^{(2 x - x^{2}) l n (x^{2} + 1)} (ln (x^{2} + 1) (2 - 2 x) + \frac{2 x ( 2 x - x ^{2} )}{x ^{2} + 1})

e^{(2 x - x^{2}) l n (x^{2} + 1)} = (x^{2} + 1)^{2 x - x^{2}}

= (ln (x^{2} + 1) (2 - 2 x) + \frac{2 x ( 2 x - x ^{2} )}{x ^{2} + 1}) (x^{2} + 1)^{2 x - x^{2}}

What is the derivative of (x^2+1^{2x-x^2}) ?
The derivative of (x^2+1^{2x-x^2}) is (ln(x^2+1)(2-2x)+(2x(2x-x^2))/(x^2+1))(x^2+1)^{2x-x^2}
What is the first derivative of (x^2+1^{2x-x^2}) ?
The first derivative of (x^2+1^{2x-x^2}) is (ln(x^2+1)(2-2x)+(2x(2x-x^2))/(x^2+1))(x^2+1)^{2x-x^2}