Question

find the derivative of ( f(x) ).

( f(x)=8^{x}-e^{5 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 \(-e^{5x}\)

Use the chain - rule \(\frac{d}{dx}(e^{u(x)}) = e^{u(x)}\cdot u^{\prime}(x)\). Let \(u = 5x\), then \(u^{\prime}=5\). So, \(\frac{d}{dx}(-e^{5x})=-e^{5x}\cdot5=-5e^{5x}\).

Step3: Combine the derivatives

Since \(f(x)=8^x - e^{5x}\), by the sum - rule \(\frac{d}{dx}(f(x)+g(x))=\frac{d}{dx}(f(x))+\frac{d}{dx}(g(x))\), we have \(f^{\prime}(x)=8^x\ln 8-5e^{5x}\).

Answer:

\(8^x\ln 8 - 5e^{5x}\)