find the derivative of ( f(x) ).\n\n( f(x)=8 e^{x}+7 cdot 4^{x} )\n\n( f^{prime}(x)= )
find the derivative of ( f(x) ).\n\n( f(x)=8 e^{x}+7 cdot 4^{x} )\n\n( f^{prime}(x)= )
Answer
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)