To find the derivative of the function [tex]\( y = (x^2 + 2x - 1)^5 \)[/tex] with respect to [tex]\( x \)[/tex], we will use the chain rule. The chain rule is used to differentiate composite functions. Let's proceed step by step.
1. Identify the outer function and the inner function:
- The outer function is [tex]\( u^5 \)[/tex], where [tex]\( u \)[/tex] is some function of [tex]\( x \)[/tex].
- The inner function is [tex]\( u = x^2 + 2x - 1 \)[/tex].
2. Differentiate the outer function with respect to the inner function [tex]\( u \)[/tex]:
- The derivative of [tex]\( u^5 \)[/tex] with respect to [tex]\( u \)[/tex] is [tex]\( 5u^4 \)[/tex].
3. Differentiate the inner function with respect to [tex]\( x \)[/tex]:
- The inner function is [tex]\( u = x^2 + 2x - 1 \)[/tex].
- The derivative of [tex]\( u \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[
\frac{d}{dx}(x^2 + 2x - 1) = 2x + 2
\][/tex]
4. Combine the results using the chain rule:
- According to the chain rule, the derivative of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
\][/tex]
5. Substitute the expressions we found:
- We have:
[tex]\[
\frac{dy}{du} = 5u^4
\][/tex]
[tex]\[
\frac{du}{dx} = 2x + 2
\][/tex]
- Therefore,
[tex]\[
\frac{dy}{dx} = 5u^4 \cdot (2x + 2)
\][/tex]
6. Substitute back the inner function [tex]\( u \)[/tex] in place of [tex]\( u \)[/tex]:
- Recall that [tex]\( u = x^2 + 2x - 1 \)[/tex].
- So,
[tex]\[
\frac{dy}{dx} = 5(x^2 + 2x - 1)^4 \cdot (2x + 2)
\][/tex]
7. Simplify the expression:
- Factor out the common factor in [tex]\( (2x + 2) \)[/tex]:
[tex]\[
2x + 2 = 2(x + 1)
\][/tex]
- Substituting this in, we get:
[tex]\[
\frac{dy}{dx} = 5(x^2 + 2x - 1)^4 \cdot 2(x + 1)
\][/tex]
- Simplify the constants:
[tex]\[
\frac{dy}{dx} = 10(x^2 + 2x - 1)^4 \cdot (x + 1)
\][/tex]
8. Rewrite the final expression neatly:
- Therefore, the derivative of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[
\frac{dy}{dx} = (10x + 10)(x^2 + 2x - 1)^4
\][/tex]
So, the derivative of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex] for [tex]\( y = (x^2 + 2x - 1)^5 \)[/tex] is [tex]\( \frac{dy}{dx} = (10x + 10)(x^2 + 2x - 1)^4 \)[/tex].
