What is the derivative of 5^{ln(x}) ?

The derivative of 5^{ln(x}) is ln(5)x^{ln(5)-1}

What is the first derivative of 5^{ln(x}) ?

The first derivative of 5^{ln(x}) is ln(5)x^{ln(5)-1}

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\frac{d}{d x} (5^{l n (x)})

ln (5) x^{l n (5) - 1}

Solution steps

\frac{d}{d x} (5^{l n (x)})

Apply exponent rule:

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

5^{l n (x)} = e^{l n (x) l n (5)}

= \frac{d}{d x} (e^{l n (x) l n (5)})

Apply the chain rule:

e^{l n (x) l n (5)} \frac{d}{d x} (ln (x) ln (5))

= e^{l n (x) l n (5)} \frac{d}{d x} (ln (x) ln (5))

\frac{d}{d x} (ln (x) ln (5)) = \frac{ln ( 5 )}{x}

= e^{l n (x) l n (5)} \frac{ln ( 5 )}{x}

Simplify

e^{l n (x) l n (5)} \frac{ln ( 5 )}{x} : ln (5) x^{l n (5) - 1}

= ln (5) x^{l n (5) - 1}

What is the derivative of 5^{ln(x}) ?
The derivative of 5^{ln(x}) is ln(5)x^{ln(5)-1}
What is the first derivative of 5^{ln(x}) ?
The first derivative of 5^{ln(x}) is ln(5)x^{ln(5)-1}