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 follow these steps:

### Step 1: Understand the function
The function given is \( f(x) = \log_e(2x) \). We need to differentiate this function with respect to \( t \).

### Step 2: Apply the chain rule
Since \( f \) is a function of \( x \) and \( x \) may be a function of \( t \), we will use the chain rule for differentiation. The chain rule states that if \( f \) is a function of \( y \) and \( y \) is a function of \( t \), then:
\[ \frac{df}{dt} = \frac{df}{dy} \cdot \frac{dy}{dt} \]

### Step 3: Differentiate \( f \) with respect to \( y \)
Let \( y = 2x \). Then, we can rewrite \( f \) as:
\[ f(y) = \log_e(y) \]
The derivative of \( f \) with respect to \( y \) is:
\[ \frac{df}{dy} = \frac{1}{y} \]

### Step 4: Find \( \frac{dy}{dt} \)
Now, we need to differentiate \( y = 2x \) with respect to \( t \):
\[ \frac{dy}{dt} = 2 \frac{dx}{dt} \]

### Step 5: Combine the results
Now, substituting back into our chain rule expression:
\[ \frac{df}{dt} = \frac{df}{dy} \cdot \frac{dy}{dt} = \frac{1}{y} \cdot (2 \frac{dx}{dt}) \]
Substituting \( y = 2x \):
\[ \frac{df}{dt} = \frac{1}{2x} \cdot (2 \frac{dx}{dt}) \]

### Step 6: Simplify the expression
The \( 2 \) in the numerator and denominator cancels out:
\[ \frac{df}{dt} = \frac{1}{x} \cdot \frac{dx}{dt} \]

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