To find the derivative [tex]\(\frac{dy}{dx}\)[/tex] of the function [tex]\( y = \left(x^2 + 2x - 1\right)^5 \)[/tex], we will use the chain rule. The chain rule allows us to differentiate composite functions.
Here is a detailed step-by-step solution:
1. Identify the outer and inner functions:
The function can be thought of as a composition of two functions. Let
[tex]\[
u = x^2 + 2x - 1
\][/tex]
and hence,
[tex]\[
y = u^5
\][/tex]
2. Differentiate the outer function with respect to [tex]\(u\)[/tex]:
The outer function is [tex]\(y = u^5\)[/tex]. Using the power rule, the derivative of [tex]\(y\)[/tex] with respect to [tex]\(u\)[/tex] is:
[tex]\[
\frac{d y}{d u} = 5u^4
\][/tex]
3. Differentiate the inner function with respect to [tex]\(x\)[/tex]:
The inner function is [tex]\(u = x^2 + 2x - 1\)[/tex]. Differentiating [tex]\(u\)[/tex] with respect to [tex]\(x\)[/tex] gives:
[tex]\[
\frac{d u}{d x} = 2x + 2
\][/tex]
4. Apply the chain rule:
The chain rule states that:
[tex]\[
\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}
\][/tex]
Substituting in the derivatives found in steps 2 and 3:
[tex]\[
\frac{d y}{d x} = 5u^4 \cdot (2x + 2)
\][/tex]
5. Substitute back [tex]\(u = x^2 + 2x - 1\)[/tex]:
Replace [tex]\(u\)[/tex] with the original expression [tex]\(x^2 + 2x - 1\)[/tex]:
[tex]\[
\frac{d y}{d x} = 5(x^2 + 2x - 1)^4 \cdot (2x + 2)
\][/tex]
6. Simplify the expression:
Factor out common terms in the derivative expression:
[tex]\[
\frac{d y}{d x} = (10x + 10)(x^2 + 2x - 1)^4
\][/tex]
Thus, the derivative of [tex]\( y = \left(x^2 + 2x - 1\right)^5 \)[/tex] with respect to [tex]\(x\)[/tex] is:
[tex]\[
\frac{dy}{dx} = (10x + 10)(x^2 + 2x - 1)^4
\][/tex]
