Suppose, the function `f(x)-f(2x)` has the derivative 5 at `x=1` and derivative 7 at `x=2`. The derivative of the function `f(x)-f(4x)` at `x=1`, has the value `10+lambda`, then the value of `lambda` is equal to……..

Answer

To solve the problem step by step, we will analyze the given information and derive the required values.

### Step 1: Define the functions
Let \( y(x) = f(x) - f(2x) \). We know that the derivative \( y'(x) \) is given by:
\[ y'(x) = f'(x) - f'(2x) \cdot 2 \]

### Step 2: Use the given derivative at \( x = 1 \)
At \( x = 1 \), we have:
\[ y'(1) = f'(1) - 2f'(2) = 5 \]
This gives us our first equation:
\[ f'(1) - 2f'(2) = 5 \quad \text{(Equation 1)} \]

### Step 3: Use the given derivative at \( x = 2 \)
At \( x = 2 \), we have:
\[ y'(2) = f'(2) - 2f'(4) = 7 \]
This gives us our second equation:
\[ f'(2) - 2f'(4) = 7 \quad \text{(Equation 2)} \]

### Step 4: Define the new function
Now, let’s define another function:
\[ h(x) = f(x) - f(4x) \]
The derivative \( h'(x) \) is given by:
\[ h'(x) = f'(x) - f'(4x) \cdot 4 \]

### Step 5: Evaluate \( h'(1) \)
At \( x = 1 \), we have:
\[ h'(1) = f'(1) - 4f'(4) \]
We know from the problem statement that:
\[ h'(1) = 10 + \lambda \]

### Step 6: Substitute \( f'(1) \) from Equation 1
From Equation 1, we can express \( f'(1) \):
\[ f'(1) = 5 + 2f'(2) \]
Substituting this into the expression for \( h'(1) \):
\[ h'(1) = (5 + 2f'(2)) - 4f'(4) = 10 + \lambda \]

### Step 7: Rearranging the equation
Now we rearrange:
\[ 5 + 2f'(2) - 4f'(4) = 10 + \lambda \]
This simplifies to:
\[ 2f'(2) - 4f'(4) = 5 + \lambda \]

### Step 8: Use Equation 2 to express \( f'(4) \)
From Equation 2, we can express \( f'(4) \):
\[ f'(4) = \frac{f'(2) - 7}{2} \]
Substituting this into the equation:
\[ 2f'(2) - 4\left(\frac{f'(2) - 7}{2}\right) = 5 + \lambda \]
This simplifies to:
\[ 2f'(2) - 2(f'(2) - 7) = 5 + \lambda \]
\[ 2f'(2) - 2f'(2) + 14 = 5 + \lambda \]
\[ 14 = 5 + \lambda \]

### Step 9: Solve for \( \lambda \)
Now, solving for \( \lambda \):
\[ \lambda = 14 - 5 = 9 \]

### Final Answer
Thus, the value of \( \lambda \) is:
\[ \boxed{9} \]