What is the derivative of e^{x^3ln(x}) ?

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

What is the first derivative of e^{x^3ln(x}) ?

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

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\frac{d}{d x} (e^{x^{3} l n (x)})

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

Solution steps

\frac{d}{d x} (e^{x^{3} l n (x)})

Simplify

e^{x^{3} l n (x)} : x^{x^{3}}

= \frac{d}{d x} (x^{x^{3}})

Apply exponent rule:

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

x^{x^{3}} = e^{x^{3} l n (x)}

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

Apply the chain rule:

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

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

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

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

e^{x^{3} l n (x)} = x^{x^{3}}

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

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