What is the derivative of (sqrt(5)^x) ?

The derivative of (sqrt(5)^x) is 1/2 ln(5)e^{(ln(5)x)/2}

What is the first derivative of (sqrt(5)^x) ?

The first derivative of (sqrt(5)^x) is 1/2 ln(5)e^{(ln(5)x)/2}

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

\frac{1}{2} ln (5) e^{\frac{l n ( 5 ) x}{2}}

Solution steps

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

Simplify

(5)^{x} : 5^{\frac{x}{2}}

= \frac{d}{d x} (5^{\frac{x}{2}})

Apply exponent rule:

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

5^{\frac{x}{2}} = e^{\frac{x}{2} l n (5)}

= \frac{d}{d x} (e^{\frac{x}{2} l n (5)})

Apply the chain rule:

e^{\frac{x}{2} l n (5)} \frac{d}{d x} (\frac{x}{2} ln (5))

= e^{\frac{x}{2} l n (5)} \frac{d}{d x} (\frac{x}{2} ln (5))

\frac{d}{d x} (\frac{x}{2} ln (5)) = \frac{1}{2} ln (5)

= e^{\frac{x}{2} l n (5)} \frac{1}{2} ln (5)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1}{2} ln (5) e^{\frac{l n ( 5 ) x}{2}}

What is the derivative of (sqrt(5)^x) ?
The derivative of (sqrt(5)^x) is 1/2 ln(5)e^{(ln(5)x)/2}
What is the first derivative of (sqrt(5)^x) ?
The first derivative of (sqrt(5)^x) is 1/2 ln(5)e^{(ln(5)x)/2}