To find the second derivative of the given function [tex]\( f(x) = 7e^{-x} - 5e^{-7x} \)[/tex], we need to follow a series of steps involving differentiation. Let's go through each step in detail.
### Step 1: Find the First Derivative
First, we need to compute the first derivative [tex]\( f'(x) \)[/tex].
Given:
[tex]\[ f(x) = 7e^{-x} - 5e^{-7x} \][/tex]
To differentiate this, we will use the chain rule for each term:
- The derivative of [tex]\( 7e^{-x} \)[/tex] is [tex]\( 7 \cdot (-1) \cdot e^{-x} = -7e^{-x} \)[/tex].
- The derivative of [tex]\( -5e^{-7x} \)[/tex] is [tex]\( -5 \cdot (-7) \cdot e^{-7x} = 35e^{-7x} \)[/tex].
So, the first derivative [tex]\( f'(x) \)[/tex] is:
[tex]\[ f'(x) = -7e^{-x} + 35e^{-7x} \][/tex]
### Step 2: Find the Second Derivative
Next, we compute the second derivative [tex]\( f''(x) \)[/tex] by differentiating [tex]\( f'(x) \)[/tex] again.
[tex]\[ f'(x) = -7e^{-x} + 35e^{-7x} \][/tex]
Again, applying the chain rule for each term:
- The derivative of [tex]\( -7e^{-x} \)[/tex] is [tex]\( -7 \cdot (-1) \cdot e^{-x} = 7e^{-x} \)[/tex].
- The derivative of [tex]\( 35e^{-7x} \)[/tex] is [tex]\( 35 \cdot (-7) \cdot e^{-7x} = -245e^{-7x} \)[/tex].
So, the second derivative [tex]\( f''(x) \)[/tex] is:
[tex]\[ f''(x) = 7e^{-x} - 245e^{-7x} \][/tex]
### Final Answer
Therefore, the second derivative [tex]\( f''(x) \)[/tex] of the function [tex]\( f(x) = 7e^{-x} - 5e^{-7x} \)[/tex] is:
[tex]\[ f''(x) = 7e^{-x} - 245e^{-7x} \][/tex]
