Question

find the derivative of ( f(x) ).

( f(x)=-5 cdot 9^{x}+3 e^{-x} )

( f^{prime}(x)= )

Explanation:

Step1: Differentiate \(-5\cdot9^{x}\)

Use the formula \((a^{x})^\prime=a^{x}\ln a\). For \(y = - 5\cdot9^{x}\), by the constant - multiple rule \((cf(x))^\prime=c\cdot f^\prime(x)\) (where \(c=-5\) and \(f(x)=9^{x}\)), we have \(y^\prime=-5\cdot9^{x}\ln9\).

Step2: Differentiate \(3e^{-x}\)

Use the formula \((e^{u(x)})^\prime = e^{u(x)}\cdot u^\prime(x)\). Let \(u(x)=-x\), then \(u^\prime(x)=-1\). For \(y = 3e^{-x}\), by the constant - multiple rule \((cf(x))^\prime=c\cdot f^\prime(x)\) (where \(c = 3\) and \(f(x)=e^{-x}\)), we have \(y^\prime=3\cdot e^{-x}\cdot(-1)=-3e^{-x}\).

Step3: Combine the derivatives

By the sum rule \((f(x)+g(x))^\prime=f^\prime(x)+g^\prime(x)\), if \(f(x)=-5\cdot9^{x}\) and \(g(x)=3e^{-x}\), then \(f^\prime(x)+g^\prime(x)=-5\cdot9^{x}\ln9-3e^{-x}\).

Answer:

\(-5\cdot9^{x}\ln9 - 3e^{-x}\)