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### How do you differentiate 2 to the power x?

let y=2x                 {take natural logs of both sides}

ln y = ln(2x)          {use rules of logs to change right hand side}

lny = xln2              {differentiate implicitly}

1/y dy/dx = ln2    {make dy/dx the subject}

dy/ dx       = y ln2  {write y in terms of x)

dy/dx = 2x . ln2

Therefore derivative of 2 to the power of x is 2x . ln2

This can be generalised as the derivative of a to the power of x (where a is a constant, a>0)  is  ax lna

3 years ago

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