Question

find the derivative of ( f(x) ).

( f(x)=8 e^{x}+7 cdot 4^{x} )

( f^{prime}(x)= )

Explanation:

Step1: Differentiate \(8e^{x}\)

The derivative of \(e^{x}\) is \(e^{x}\). Using the constant - multiple rule \((cf(x))^\prime = cf^\prime(x)\), for \(y = 8e^{x}\), we have \((8e^{x})^\prime=8e^{x}\).

Step2: Differentiate \(7\cdot4^{x}\)

The formula for the derivative of \(a^{x}\) is \((a^{x})^\prime=a^{x}\ln a\). Using the constant - multiple rule \((cf(x))^\prime = cf^\prime(x)\) with \(c = 7\) and \(a = 4\), we get \((7\cdot4^{x})^\prime=7\cdot4^{x}\ln4\).

Step3: Use the sum rule

The sum rule of differentiation states that \((u + v)^\prime=u^\prime+v^\prime\). If \(u = 8e^{x}\) and \(v = 7\cdot4^{x}\), then \(f^\prime(x)=(8e^{x}+7\cdot4^{x})^\prime=(8e^{x})^\prime+(7\cdot4^{x})^\prime\).

Answer:

\(8e^{x}+7\cdot4^{x}\ln4\)