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

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)