Question

find the derivative of ( f(x) ).

( f(x)=e^{x}-10^{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 \(10^x\)

Use the formula \(\frac{d}{dx}(a^x)=a^x\ln a\). For \(a = 10\), \(\frac{d}{dx}(10^x)=10^x\ln 10\).

Step3: Apply the difference rule

If \(f(x)=u(x)-v(x)\), then \(f^{\prime}(x)=u^{\prime}(x)-v^{\prime}(x)\). Here \(u(x)=e^x\), \(u^{\prime}(x)=e^x\) and \(v(x)=10^x\), \(v^{\prime}(x)=10^x\ln 10\). So \(f^{\prime}(x)=e^x - 10^x\ln 10\).

Answer:

\(e^x-10^x\ln 10\)