What is the derivative of 2^{e^x} ?

The derivative of 2^{e^x} is ln(2)*2^{e^x}e^x

What is the first derivative of 2^{e^x} ?

The first derivative of 2^{e^x} is ln(2)*2^{e^x}e^x

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

ln (2) \cdot 2^{e^{x}} e^{x}

Solution steps

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

Apply exponent rule:

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

2^{e^{x}} = e^{e^{x} l n (2)}

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

Apply the chain rule:

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

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

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

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

Simplify

e^{e^{x} l n (2)} ln (2) e^{x} : ln (2) \cdot 2^{e^{x}} e^{x}

= ln (2) \cdot 2^{e^{x}} e^{x}

What is the derivative of 2^{e^x} ?
The derivative of 2^{e^x} is ln(2)*2^{e^x}e^x
What is the first derivative of 2^{e^x} ?
The first derivative of 2^{e^x} is ln(2)*2^{e^x}e^x