To find the derivative of [tex]\( f(x) = (6 x^4 + 2 x^3)(5 x - 10 x^2 + 18 x^5 - 4) \)[/tex], we will use the product rule of differentiation. The product rule states that if we have two functions [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex], then the derivative of their product is given by:
[tex]\[ (u(x) \cdot v(x))' = u'(x) \cdot v(x) + u(x) \cdot v'(x) \][/tex]
Let's denote:
[tex]\[ u(x) = 6 x^4 + 2 x^3 \][/tex]
[tex]\[ v(x) = 5 x - 10 x^2 + 18 x^5 - 4 \][/tex]
First, we need to find the derivatives of [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex].
### Step 1: Find [tex]\( u'(x) \)[/tex]
[tex]\[ u(x) = 6 x^4 + 2 x^3 \][/tex]
Differentiating [tex]\( u(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ u'(x) = \frac{d}{d x}(6 x^4) + \frac{d}{d x}(2 x^3) \][/tex]
[tex]\[ u'(x) = 24 x^3 + 6 x^2 \][/tex]
### Step 2: Find [tex]\( v'(x) \)[/tex]
[tex]\[ v(x) = 5 x - 10 x^2 + 18 x^5 - 4 \][/tex]
Differentiating [tex]\( v(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ v'(x) = \frac{d}{d x}(5 x) + \frac{d}{d x}(-10 x^2) + \frac{d}{d x}(18 x^5) + \frac{d}{d x}(-4) \][/tex]
[tex]\[ v'(x) = 5 - 20 x + 90 x^4 \][/tex]
### Step 3: Apply the Product Rule
Using the product rule, we have:
[tex]\[ f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) \][/tex]
Substituting [tex]\( u(x) \)[/tex], [tex]\( u'(x) \)[/tex], [tex]\( v(x) \)[/tex], and [tex]\( v'(x) \)[/tex]:
[tex]\[ f'(x) = (24 x^3 + 6 x^2)(5 x - 10 x^2 + 18 x^5 - 4) + (6 x^4 + 2 x^3)(5 - 20 x + 90 x^4) \][/tex]
### Conclusion:
Therefore, the derivative of the given function is:
[tex]\[ f'(x) = (24 x^3 + 6 x^2)(5 x - 10 x^2 + 18 x^5 - 4) + (6 x^4 + 2 x^3)(5 - 20 x + 90 x^4) \][/tex]
This is the step-by-step solution to finding the derivative of the function [tex]\( f(x) \)[/tex].
