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

Explanation:

Step1: Differentiate (e^{-x})

Use the chain rule. Let (u = -x), then (\frac{d}{dx}(e^u)=e^u\cdot\frac{du}{dx}). Here (\frac{du}{dx}=- 1), so (\frac{d}{dx}(e^{-x})=e^{-x}\cdot(-1)=-e^{-x})

Step2: Differentiate (2^{x})

Use the formula (\frac{d}{dx}(a^{x})=a^{x}\ln a). For (a = 2), (\frac{d}{dx}(2^{x})=2^{x}\ln2)

Step3: Combine the derivatives

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

Answer:

(-e^{-x}-2^{x}\ln2)