What is the derivative of x^{ax} ?

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

What is the first derivative of x^{ax} ?

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

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\frac{d}{d x} (x^{a x})

a x^{a x} (ln (x) + 1)

Solution steps

\frac{d}{d x} (x^{a x})

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

\frac{d}{d x} (a x ln (x)) = a (ln (x) + 1)

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

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

= a x^{a x} (ln (x) + 1)

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