What is the derivative of (2/5 ^{sqrt(x)}) ?

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

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

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

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

\frac{ln ( \frac{2}{5} ) \cdot 2 ^{x - 1}}{5 ^{x} x}

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

= \frac{ln ( \frac{2}{5} ) \cdot 2 ^{x - 1}}{5 ^{x} x}

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