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Find the derivative of 2x2

Find the derivative of the following function:

    \[ f(x) = 2^{x^2}. \]


First, we use the definition of the exponential to rewrite this as,

    \[ f(x) = 2^{x^2} = e^{\log (2^{x^2})} = e^{x^2 \log 2}. \]

Then, we can take the derivative of this,

    \[ f'(x) = (2x \log 2) \cdot (e^{x^2 \log 2}) = (2x \log 2)(2^{x^2}) = 2^{x^2+1} x \log 2. \]

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