Questions: If f(x) = 5(2^x), find f'(x). Select the correct answer below: f'(x) = 2^x ln x f'(x) = 2^x(2 ln 5) f'(x) = 2^x(5 ln 2) f'(x) = 5(2^x) f'(x) = (5 x) 2^(x-1)

Transcript text: If $f(x)=5\left(2^{x}\right)$, find $f^{\prime}(x)$. Select the correct answer below: $f^{\prime}(x)=2^{x} \ln x$ $f^{\prime}(x)=2^{x}(2 \ln 5)$ $f^{\prime}(x)=2^{x}(5 \ln 2)$ $f^{\prime}(x)=5\left(2^{x}\right)$ $f^{\prime}(x)=(5 x) 2^{x-1}$

Solution

Solution Steps

To find the derivative of the function $ f(x) = 5 \left(2^x\right) $, we need to use the chain rule and the fact that the derivative of $ 2^x $ is $ 2^x \ln(2) $.

Step 1: Define the Function

We start with the function defined as: \[ f(x) = 5 \cdot 2^x \]

Step 2: Differentiate the Function

To find the derivative $ f'(x) $, we apply the chain rule. The derivative of $ 2^x $ is $ 2^x \ln(2) $. Therefore, we have: \[ f'(x) = 5 \cdot \frac{d}{dx}(2^x) = 5 \cdot 2^x \ln(2) \]

Step 3: Simplify the Derivative

Thus, the derivative simplifies to: \[ f'(x) = 5 \cdot 2^x \ln(2) \]