Question

find the derivative of ( f(x) ).

( f(x)=e^{x}-2^{x} )

( f^{prime}(x)= )

Explanation:

Step1: Differentiate \(e^x\)

The derivative of \(e^x\) with respect to \(x\) is \(e^x\), i.e., \(\frac{d}{dx}(e^x)=e^x\).

Step2: Differentiate \(2^x\)

Using the formula \(\frac{d}{dx}(a^x)=a^x\ln a\) (where \(a = 2\)), we get \(\frac{d}{dx}(2^x)=2^x\ln 2\).

Step3: Apply the difference rule

Since \(f(x)=e^x - 2^x\), by the difference rule \((u - v)'=u' - v'\) (where \(u = e^x\) and \(v = 2^x\)), we have \(f'(x)=\frac{d}{dx}(e^x)-\frac{d}{dx}(2^x)\).

Substituting the derivatives from Step1 and Step2: \(f'(x)=e^x-2^x\ln 2\).

Answer:

\(e^x - 2^x\ln 2\)