Sure, let's go through the solution step-by-step.
Given the function:
[tex]\[ f(x) = (5x + 5)^{-4} \][/tex]
First, we need to find the derivative [tex]\( f'(x) \)[/tex].
### Step 1: Find the Derivative [tex]\( f'(x) \)[/tex]
We'll use the chain rule for differentiation. The chain rule states that if you have a composite function [tex]\( f(g(x)) \)[/tex], then its derivative is [tex]\( f'(g(x)) \cdot g'(x) \)[/tex].
In our case, we can set:
[tex]\[ g(x) = 5x + 5 \][/tex]
and
[tex]\[ h(x) = g(x)^{-4} \][/tex]
To differentiate [tex]\( f(x) \)[/tex], we first differentiate [tex]\( h(y) = y^{-4} \)[/tex]:
[tex]\[ h'(y) = -4y^{-5} \][/tex]
Then by the chain rule:
[tex]\[ f'(x) = h'(g(x)) \cdot g'(x) \][/tex]
We find [tex]\( g'(x) \)[/tex]:
[tex]\[ g'(x) = \frac{d}{dx}(5x + 5) = 5 \][/tex]
Therefore:
[tex]\[ f'(x) = -4(5x + 5)^{-5} \cdot 5 \][/tex]
[tex]\[ f'(x) = -20(5x + 5)^{-5} \][/tex]
So the derivative is:
[tex]\[ f'(x) = \frac{-20}{(5x + 5)^5} \][/tex]
### Step 2: Evaluate [tex]\( f'(4) \)[/tex]
To find [tex]\( f'(4) \)[/tex], simply substitute [tex]\( x = 4 \)[/tex] into the derivative:
[tex]\[ f'(4) = \frac{-20}{(5 \cdot 4 + 5)^5} \][/tex]
[tex]\[ = \frac{-20}{(20 + 5)^5} \][/tex]
[tex]\[ = \frac{-20}{25^5} \][/tex]
Calculate [tex]\( 25^5 \)[/tex]:
[tex]\[ 25^5 = 25 \cdot 25 \cdot 25 \cdot 25 \cdot 25 = 9765625 \][/tex]
Therefore:
[tex]\[ f'(4) = \frac{-20}{9765625} \][/tex]
Approximating this value gives:
[tex]\[ f'(4) \approx -2.048 \times 10^{-6} \][/tex]
So, the derivative function is:
[tex]\[ f'(x) = \frac{-20}{(5x + 5)^5} \][/tex]
And the value of the derivative at [tex]\( x = 4 \)[/tex] is:
[tex]\[ f'(4) \approx -2.048 \times 10^{-6} \][/tex]
