Certainly! To find the derivative [tex]\( \frac{dy}{dx} \)[/tex] of the function [tex]\( y = \left(x^4 + 3x\right)^6 \)[/tex], we will use the chain rule, which is essential when dealing with composite functions. Here's a step-by-step solution:
1. Identify the outer and inner functions:
- The outer function is [tex]\( u^6 \)[/tex], where [tex]\( u = x^4 + 3x \)[/tex].
- The inner function is [tex]\( u = x^4 + 3x \)[/tex].
2. Differentiate the outer function with respect to the inner function:
- The outer function in terms of [tex]\( u \)[/tex] is [tex]\( u^6 \)[/tex].
- The derivative of [tex]\( u^6 \)[/tex] with respect to [tex]\( u \)[/tex] is [tex]\( 6u^5 \)[/tex].
3. Differentiate the inner function with respect to [tex]\( x \)[/tex]:
- The inner function is [tex]\( u = x^4 + 3x \)[/tex].
- The derivative of [tex]\( x^4 + 3x \)[/tex] with respect to [tex]\( x \)[/tex] is [tex]\( 4x^3 + 3 \)[/tex].
4. Apply the chain rule:
- The chain rule states that [tex]\( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \)[/tex].
- Substitute the inner function's derivative and the outer function's derivative into the chain rule:
[tex]\[
\frac{dy}{dx} = 6u^5 \cdot (4x^3 + 3)
\][/tex]
- Substitute back [tex]\( u = x^4 + 3x \)[/tex]:
[tex]\[
\frac{dy}{dx} = 6(x^4 + 3x)^5 \cdot (4x^3 + 3)
\][/tex]
5. Simplify the expression:
- Factor out common terms:
[tex]\[
\frac{dy}{dx} = (6(4x^3 + 3))(x^4 + 3x)^5
\][/tex]
- Simplify within the parentheses:
[tex]\[
6(4x^3 + 3) = 24x^3 + 18
\][/tex]
- Therefore, the derivative becomes:
[tex]\[
\frac{dy}{dx} = (24x^3 + 18)(x^4 + 3x)^5
\][/tex]
Hence, the derivative of the function [tex]\( y = \left( x^4 + 3x \right)^6 \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[
\boxed{(24x^3 + 18)(x^4 + 3x)^5}
\][/tex]
