Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.

Find [tex]\frac{d y}{d x}[/tex] when

(i) [tex]y=\left(x^2+2x-1\right)^5[/tex]


Sagot :

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]
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.