Question

find the derivative of $f(x)$.
$f(x)=e^{-2x}+5^{x}$
$f(x)=$

Explanation:

Step1: Differentiate \(e^{-2x}\)

Use the chain rule. Let \(u = -2x\), then \(\frac{d}{dx}(e^u)=e^u\cdot\frac{du}{dx}\).
\(\frac{d}{dx}(e^{-2x})=e^{-2x}\cdot(-2)=-2e^{-2x}\)

Step2: Differentiate \(5^x\)

Use the formula \(\frac{d}{dx}(a^x)=a^x\ln a\) (here \(a = 5\)).
\(\frac{d}{dx}(5^x)=5^x\ln 5\)

Step3: Sum the derivatives

Since \(f(x)=e^{-2x}+5^x\), by the sum rule \((u + v)'=u'+v'\) (where \(u = e^{-2x}\) and \(v = 5^x\)).
\(f'(x)=\frac{d}{dx}(e^{-2x})+\frac{d}{dx}(5^x)\)

Answer:

\(-2e^{-2x}+5^x\ln 5\)