find the derivative of ( f(x) ).\n\n( f(x)=8^{x}-e^{5 x} )\n( 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})