What is the derivative of (x+1^{x^2+x^3}) ?

The derivative of (x+1^{x^2+x^3}) is (ln(x+1)(2x+3x^2)+x^2)(x+1)^{x^2+x^3}

What is the first derivative of (x+1^{x^2+x^3}) ?

The first derivative of (x+1^{x^2+x^3}) is (ln(x+1)(2x+3x^2)+x^2)(x+1)^{x^2+x^3}

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\frac{d}{d x} ((x + 1)^{x^{2} + x^{3}})

(ln (x + 1) (2 x + 3 x^{2}) + x^{2}) (x + 1)^{x^{2} + x^{3}}

Solution steps

\frac{d}{d x} ((x + 1)^{x^{2} + x^{3}})

Apply exponent rule:

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

(x + 1)^{x^{2} + x^{3}} = e^{(x^{2} + x^{3}) l n (x + 1)}

= \frac{d}{d x} (e^{(x^{2} + x^{3}) l n (x + 1)})

Apply the chain rule:

e^{(x^{2} + x^{3}) l n (x + 1)} \frac{d}{d x} ((x^{2} + x^{3}) ln (x + 1))

= e^{(x^{2} + x^{3}) l n (x + 1)} \frac{d}{d x} ((x^{2} + x^{3}) ln (x + 1))

\frac{d}{d x} ((x^{2} + x^{3}) ln (x + 1)) = ln (x + 1) (2 x + 3 x^{2}) + x^{2}

= e^{(x^{2} + x^{3}) l n (x + 1)} (ln (x + 1) (2 x + 3 x^{2}) + x^{2})

e^{(x^{2} + x^{3}) l n (x + 1)} = (x + 1)^{x^{2} + x^{3}}

= (ln (x + 1) (2 x + 3 x^{2}) + x^{2}) (x + 1)^{x^{2} + x^{3}}

\int e^{8 x} d x

t y^{^{'}} - y = 3 t^{2} y^{2}

2 x^{3} + 3 x^{2} - 12 x + 6

\int \frac{1}{x 225 x ^{2} - 225} d x

What is the derivative of (x+1^{x^2+x^3}) ?
The derivative of (x+1^{x^2+x^3}) is (ln(x+1)(2x+3x^2)+x^2)(x+1)^{x^2+x^3}
What is the first derivative of (x+1^{x^2+x^3}) ?
The first derivative of (x+1^{x^2+x^3}) is (ln(x+1)(2x+3x^2)+x^2)(x+1)^{x^2+x^3}