Question

find the derivative of $f(x)$.
$f(x)=e^{x}-4^{x}$
$f^{prime}(x)=$

Explanation:

Step1: Differentiate \(e^x\)

The derivative of \(e^x\) with respect to \(x\) is \(e^x\).

Step2: Differentiate \(4^x\)

Using the formula \(\frac{d}{dx}(a^x)=a^x\ln a\) (where \(a = 4\)), the derivative of \(4^x\) is \(4^x\ln 4\).

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 \(v(x)=4^x\), \(v^\prime(x)=4^x\ln 4\). So \(f^\prime(x)=e^x - 4^x\ln 4\).

Answer:

\(e^x-4^x\ln 4\)