To differentiate the function [tex]\( f(x) = (x+8)^4 \sec(3x) \)[/tex], we will use a combination of differentiation rules: the product rule and the chain rule.
### Step 1: Identify the components of the function
The function [tex]\( f(x) \)[/tex] can be seen as a product of two functions:
- [tex]\( u(x) = (x+8)^4 \)[/tex]
- [tex]\( v(x) = \sec(3x) \)[/tex]
### Step 2: Differentiate [tex]\( u(x) \)[/tex]
Using the chain rule:
[tex]\[ u(x) = (x+8)^4 \][/tex]
Let [tex]\( g(x) = x + 8 \)[/tex] and [tex]\( h(g) = g^4 \)[/tex]. We have:
[tex]\[ u(x) = h(g(x)) = (x + 8)^4 \][/tex]
Differentiate [tex]\( h(g) \)[/tex] with respect to [tex]\( g \)[/tex]:
[tex]\[ h'(g) = 4g^3 \][/tex]
Then differentiate [tex]\( g(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ g'(x) = 1 \][/tex]
Now apply the chain rule:
[tex]\[ u'(x) = h'(g(x)) \cdot g'(x) = 4(x+8)^3 \cdot 1 = 4(x+8)^3 \][/tex]
### Step 3: Differentiate [tex]\( v(x) \)[/tex]
Using the chain rule:
[tex]\[ v(x) = \sec(3x) \][/tex]
Let [tex]\( k(x) = 3x \)[/tex] and [tex]\( m(k) = \sec(k) \)[/tex]. We have:
[tex]\[ v(x) = m(k(x)) = \sec(3x) \][/tex]
Differentiate [tex]\( m(k) \)[/tex] with respect to [tex]\( k \)[/tex]:
[tex]\[ m'(k) = \sec(k) \tan(k) \][/tex]
Then differentiate [tex]\( k(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ k'(x) = 3 \][/tex]
Now apply the chain rule:
[tex]\[ v'(x) = m'(k(x)) \cdot k'(x) = \sec(3x) \tan(3x) \cdot 3 = 3 \sec(3x) \tan(3x) \][/tex]
### Step 4: Apply the Product Rule
The product rule states that:
[tex]\[ (uv)' = u'v + uv' \][/tex]
Substituting [tex]\( u(x) \)[/tex], [tex]\( u'(x) \)[/tex], [tex]\( v(x) \)[/tex], and [tex]\( v'(x) \)[/tex]:
[tex]\[ f'(x) = u'(x)v(x) + u(x)v'(x) \][/tex]
[tex]\[ f'(x) = 4(x+8)^3 \sec(3x) + (x+8)^4 \cdot 3 \sec(3x) \tan(3x) \][/tex]
### Step 5: Simplify the expression
Combine like terms:
[tex]\[ f'(x) = 4(x+8)^3 \sec(3x) + 3(x+8)^4 \sec(3x) \tan(3x) \][/tex]
### Final Answer
[tex]\[ f'(x) = 3(x + 8)^4 \tan(3x) \sec(3x) + 4(x + 8)^3 \sec(3x) \][/tex]
This is the differentiated form of the given function [tex]\( f(x) = (x+8)^4 \sec(3x) \)[/tex].
