The derivative of function f(x) = `log_e(2x)` w.r.t. t is

Options

A. `f(x)=1/(2x) (dx)/(dt)`
B. `f(x)=1/x (dx)/(dt)`
C. `f(x)=1/(x^2) (dx)/(dt)`
D. `f(x)=1/(2x^2) (dx)/(dt)`

Answer

To find the derivative of the function \( f(x) = \log_e(2x) \) with respect to \( t \), we will use the chain rule. Here are the steps to solve the problem:

### Step 1: Identify the function to differentiate
We have the function:
\[ f(x) = \log_e(2x) \]

### Step 2: Apply the chain rule
To differentiate \( f(x) \) with respect to \( t \), we will use the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is:
\[ \frac{df}{dt} = \frac{df}{dx} \cdot \frac{dx}{dt} \]

### Step 3: Differentiate \( f(x) \) with respect to \( x \)
First, we need to differentiate \( f(x) \) with respect to \( x \):
\[ \frac{df}{dx} = \frac{d}{dx} \log_e(2x) \]
Using the logarithmic differentiation rule, we have:
\[ \frac{d}{dx} \log_e(2x) = \frac{1}{2x} \cdot \frac{d}{dx}(2x) \]
The derivative of \( 2x \) with respect to \( x \) is \( 2 \), so:
\[ \frac{df}{dx} = \frac{1}{2x} \cdot 2 = \frac{1}{x} \]

### Step 4: Multiply by \( \frac{dx}{dt} \)
Now, we multiply by \( \frac{dx}{dt} \):
\[ \frac{df}{dt} = \frac{df}{dx} \cdot \frac{dx}{dt} = \frac{1}{x} \cdot \frac{dx}{dt} \]

### Final Result
Thus, the derivative of the function \( f(x) = \log_e(2x) \) with respect to \( t \) is:
\[ \frac{df}{dt} = \frac{1}{x} \cdot \frac{dx}{dt} \]