To find the derivative [tex]\( \frac{dy}{dx} \)[/tex] when [tex]\( y = \left(x^2 + 2x - 1\right)^5 \)[/tex], we need to use the chain rule. Here's the step-by-step solution:
1. Identify the outer and inner functions:
- The outer function is [tex]\( u^5 \)[/tex], where [tex]\( u = x^2 + 2x - 1 \)[/tex].
- The inner function is [tex]\( x^2 + 2x - 1 \)[/tex].
2. Take the derivative of the outer function with respect to the inner function [tex]\( u \)[/tex]:
[tex]\[
\frac{d}{du}(u^5) = 5u^4
\][/tex]
Since [tex]\( u = x^2 + 2x - 1 \)[/tex], this becomes:
[tex]\[
5(x^2 + 2x - 1)^4
\][/tex]
3. Take the derivative of the inner function [tex]\( u \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[
\frac{d}{dx}(x^2 + 2x - 1) = 2x + 2
\][/tex]
4. Apply the chain rule:
The chain rule states that:
[tex]\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
\][/tex]
Substituting in the derivatives we found:
[tex]\[
\frac{dy}{dx} = 5(x^2 + 2x - 1)^4 \cdot (2x + 2)
\][/tex]
5. Simplify the expression:
Factor out the common terms:
[tex]\[
\frac{dy}{dx} = 5(2x + 2)(x^2 + 2x - 1)^4
\][/tex]
[tex]\[
\frac{dy}{dx} = 10(x + 1)(x^2 + 2x - 1)^4
\][/tex]
The simplified derivative is:
[tex]\[
\frac{dy}{dx} = 10(x + 1)(x^2 + 2x - 1)^4
\][/tex]
However, in its unsimplified form, it's:
[tex]\[
\frac{dy}{dx} = (10x + 10)(x^2 + 2x - 1)^4
\][/tex]
Both forms represent the same result:
[tex]\[
\left( \frac{dy}{dx} \right) = \left( (10x + 10)(x^2 + 2x - 1)^4, 10(x + 1)(x^2 + 2x - 1)^4 \right)
\][/tex]
