What is the derivative of-ln(2*2^{-x}) ?

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

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

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

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

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

Solution steps

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

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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