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

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

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

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

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

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

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