Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
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].
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].
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.
