Sure, let's find the derivative of the function [tex]\( f(x) = 5 e^{8 x^5 + 4 x^3} \)[/tex] using the chain rule. We will proceed step by step.
### Step 1: Recognize the outer and inner functions
1. Outer function: [tex]\( g(u) = 5 e^u \)[/tex], where [tex]\( u \)[/tex] is a function of [tex]\( x \)[/tex]
2. Inner function: [tex]\( u(x) = 8 x^5 + 4 x^3 \)[/tex]
### Step 2: Differentiate the outer function with respect to the inner function
First, we differentiate the outer function [tex]\( g(u) \)[/tex] with respect to [tex]\( u \)[/tex]:
[tex]\[ g'(u) = 5 e^u \][/tex]
Since the derivative of [tex]\( e^u \)[/tex] with respect to [tex]\( u \)[/tex] is [tex]\( e^u \)[/tex], we have:
[tex]\[ \frac{d}{du} \left( 5 e^u \right) = 5 e^u \][/tex]
### Step 3: Differentiate the inner function with respect to [tex]\( x \)[/tex]
Next, we differentiate the inner function [tex]\( u(x) = 8 x^5 + 4 x^3 \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ u'(x) = \frac{d}{dx} \left( 8 x^5 + 4 x^3 \right) \][/tex]
Applying the power rule to each term, we get:
[tex]\[ \frac{d}{dx} \left( 8 x^5 \right) = 40 x^4 \][/tex]
[tex]\[ \frac{d}{dx} \left( 4 x^3 \right) = 12 x^2 \][/tex]
So, the derivative of the inner function is:
[tex]\[ u'(x) = 40 x^4 + 12 x^2 \][/tex]
### Step 4: Apply the chain rule
The chain rule states that:
[tex]\[ f'(x) = g'(u) \cdot u'(x) \][/tex]
Substituting [tex]\( g'(u) \)[/tex] and [tex]\( u'(x) \)[/tex] into this formula, we get:
[tex]\[ f'(x) = 5 e^u \cdot (40 x^4 + 12 x^2) \][/tex]
### Step 5: Substitute back the inner function [tex]\( u \)[/tex]
Recall that the inner function [tex]\( u \)[/tex] is [tex]\( 8 x^5 + 4 x^3 \)[/tex]. Substitute [tex]\( u \)[/tex] back into the expression:
[tex]\[ f'(x) = 5 e^{8 x^5 + 4 x^3} \cdot (40 x^4 + 12 x^2) \][/tex]
### Final Answer
Thus, the derivative of [tex]\( f(x) \)[/tex] is:
[tex]\[ f'(x) = 5 \left( 40 x^4 + 12 x^2 \right) e^{8 x^5 + 4 x^3} \][/tex]
