Certainly! Let's find the derivative of the given function [tex]\( f(x) = 3 \left( 6 x^{10} + 8 x^8 \right)^{13} \)[/tex] using the chain rule.
### Step-by-Step Solution
1. Identify the outer and inner functions:
- Outer function: [tex]\( g(u) = 3u^{13} \)[/tex] where [tex]\( u = 6x^{10} + 8x^8 \)[/tex].
- Inner function: [tex]\( u(x) = 6x^{10} + 8x^8 \)[/tex].
2. Differentiate the outer function with respect to the inner function [tex]\( u \)[/tex]:
- The derivative of [tex]\( g(u) = 3u^{13} \)[/tex] with respect to [tex]\( u \)[/tex] is [tex]\( g'(u) = 3 \cdot 13 u^{12} = 39u^{12} \)[/tex].
3. Differentiate the inner function with respect to [tex]\( x \)[/tex]:
- The derivative of [tex]\( u(x) = 6x^{10} + 8x^8 \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[
u'(x) = \frac{d}{dx}(6x^{10}) + \frac{d}{dx}(8x^8) = 60x^9 + 64x^7
\][/tex]
4. Apply the chain rule:
- The chain rule states: [tex]\( \frac{df}{dx} = g'(u) \cdot u'(x) \)[/tex].
- Substituting the derivatives we found:
[tex]\[
f'(x) = 39u^{12} \cdot (60x^9 + 64x^7)
\][/tex]
- Recall [tex]\( u = 6x^{10} + 8x^8 \)[/tex]:
[tex]\[
f'(x) = 39 (6x^{10} + 8x^8)^{12} \cdot (60x^9 + 64x^7)
\][/tex]
5. Simplify the expression:
- Simplifying further to make it more readable:
[tex]\[
f'(x) = 39 (6x^{10} + 8x^8)^{12} (60x^9 + 64x^7)
\][/tex]
The above method can be verified by substituting back into more specialized forms. However, let's keep it at this simplified stage.
So, the derivative of [tex]\( f(x) \)[/tex] using the chain rule is:
[tex]\[
f'(x) = 39 (6x^{10} + 8x^8)^{12} (60x^9 + 64x^7)
\][/tex]
