Question

find the derivative of ( f(x) ).

( f(x)=8^{x}-e^{x} )

( f^{prime}(x)= )

Explanation:

Step1: Differentiate \(8^x\)

Use the formula \(\frac{d}{dx}(a^x)=a^x\ln a\), so \(\frac{d}{dx}(8^x)=8^x\ln 8\)

Step2: Differentiate \(-e^x\)

Use the formula \(\frac{d}{dx}(e^x)=e^x\), so \(\frac{d}{dx}(-e^x)=-e^x\)

Step3: Combine the derivatives

By the sum - difference rule \((u - v)'=u'-v'\), where \(u = 8^x\) and \(v=e^x\). Then \(f'(x)=\frac{d}{dx}(8^x)-\frac{d}{dx}(e^x)\)

Answer:

\(8^x\ln 8 - e^x\)