What is the derivative of 4^{e^x} ?

The derivative of 4^{e^x} is ln(2)*2^{1+2e^x}e^x

What is the first derivative of 4^{e^x} ?

The first derivative of 4^{e^x} is ln(2)*2^{1+2e^x}e^x

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\frac{d}{d x} (4^{e^{x}})

ln (2) \cdot 2^{1 + 2 e^{x}} e^{x}

Solution steps

\frac{d}{d x} (4^{e^{x}})

Apply exponent rule:

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

4^{e^{x}} = e^{e^{x} l n (4)}

= \frac{d}{d x} (e^{e^{x} l n (4)})

Apply the chain rule:

e^{e^{x} l n (4)} \frac{d}{d x} (e^{x} ln (4))

= e^{e^{x} l n (4)} \frac{d}{d x} (e^{x} ln (4))

\frac{d}{d x} (e^{x} ln (4)) = 2 ln (2) e^{x}

= e^{e^{x} l n (4)} \cdot 2 ln (2) e^{x}

Simplify

e^{e^{x} l n (4)} \cdot 2 ln (2) e^{x} : ln (2) \cdot 2^{1 + 2 e^{x}} e^{x}

= ln (2) \cdot 2^{1 + 2 e^{x}} e^{x}

What is the derivative of 4^{e^x} ?
The derivative of 4^{e^x} is ln(2)*2^{1+2e^x}e^x
What is the first derivative of 4^{e^x} ?
The first derivative of 4^{e^x} is ln(2)*2^{1+2e^x}e^x