Sure! Let's go through the process of finding the derivative of the function [tex]\( f(x) = (10 x^2 + 2 x + 3) (8 x^{-4} - 2 x + 3)^3 \)[/tex].
Step 1: Define the Function
Given:
[tex]\[ f(x) = (10 x^2 + 2 x + 3)(8 x^{-4} - 2 x + 3)^3 \][/tex]
Step 2: Apply the Product Rule
To differentiate [tex]\( f(x) \)[/tex], we need to use the product rule which states:
[tex]\[ (uv)' = u'v + uv' \][/tex]
Here, let
[tex]\[ u = 10 x^2 + 2 x + 3 \][/tex]
[tex]\[ v = (8 x^{-4} - 2 x + 3)^3 \][/tex]
To apply the product rule, we need to first find [tex]\( u' \)[/tex] and [tex]\( v' \)[/tex].
Step 3: Differentiate [tex]\( u \)[/tex]
[tex]\[ u = 10 x^2 + 2 x + 3 \][/tex]
[tex]\[ u' = \frac{d}{dx}(10 x^2) + \frac{d}{dx}(2 x) + \frac{d}{dx}(3) \][/tex]
[tex]\[ u' = 20 x + 2 + 0 \][/tex]
[tex]\[ u' = 20 x + 2 \][/tex]
Step 4: Differentiate [tex]\( v \)[/tex] using the Chain Rule
[tex]\[ v = (8 x^{-4} - 2 x + 3)^3 \][/tex]
Let
[tex]\[ g(x) = 8 x^{-4} - 2 x + 3 \][/tex]
Then
[tex]\[ v = [g(x)]^3 \][/tex]
Using the chain rule:
[tex]\[ v' = 3 [g(x)]^2 g'(x) \][/tex]
First, find [tex]\( g'(x) \)[/tex]:
[tex]\[ g(x) = 8 x^{-4} - 2 x + 3 \][/tex]
Differentiate term-by-term:
[tex]\[ g'(x) = \frac{d}{dx}(8 x^{-4}) + \frac{d}{dx}(-2 x) + \frac{d}{dx}(3) \][/tex]
[tex]\[ g'(x) = 8(-4)x^{-5} - 2 + 0 \][/tex]
[tex]\[ g'(x) = -32 x^{-5} - 2 \][/tex]
So,
[tex]\[ v' = 3 (8 x^{-4} - 2 x + 3)^2 (-32 x^{-5} - 2) \][/tex]
Step 5: Combine using the Product Rule
Combine [tex]\( u' \)[/tex] and [tex]\( v' \)[/tex] using the product rule:
[tex]\[ f'(x) = u' v + u v' \][/tex]
Letting:
[tex]\[ u = 10 x^2 + 2 x + 3 \][/tex]
[tex]\[ u' = 20 x + 2 \][/tex]
[tex]\[ v = (8 x^{-4} - 2 x + 3)^3 \][/tex]
[tex]\[ v' = 3 (8 x^{-4} - 2 x + 3)^2 (-32 x^{-5} - 2) \][/tex]
We get:
[tex]\[ f'(x) = (20 x + 2)(8 x^{-4} - 2 x + 3)^3 + (10 x^2 + 2 x + 3) \cdot 3 (8 x^{-4} - 2 x + 3)^2 (-32 x^{-5} - 2) \][/tex]
Step 6: Simplify the Expression
After combining and simplifying the terms, the final expression for the derivative is:
[tex]\[ f'(x) = \frac{2 (x^4 (3 - 2 x) + 8)^2 \left[ x (10 x + 1)(x^4 (3 - 2 x) + 8) - 3 (x^5 + 16) (10 x^2 + 2 x + 3) \right]}{x^{13}} \][/tex]
This is the derivation in simplified form, expressed with appropriate positive and negative exponents, and all without using radicals.
