Lesson Example Discussion Quiz: Class Homework |
Step-5 |
Title: Differentiation |
Grade: 10-a Lesson: S2-L5 |
Explanation: Hello Students, time to practice and review the steps for the problem. |
Lesson Steps
| Step | Type | Explanation | Answer |
|---|---|---|---|
1 |
Problem |
Find the derivative of the function \$"f"("x") = ("x"^2 + 1)^("x"^3 - 2"x")\$. |
|
2 |
Step |
The given function |
\$"f"("x") = ("x"^2 + 1)^("x"^3 - 2"x")\$ |
3 |
Formula: |
The power rule formula: |
\$ "d"/"dx" "x"^"n" = "n" "x"^("n" - 1) \$ |
4 |
Step |
Now we can use the chain rule to find the derivative of f(x), then simplify. |
\$"d"/"dx" "f"("x") = "d"/"dx" (("x"^2 + 1)^("x"^3 - 2"x"))\$ \$ "f"'("x") = ("x"^3 - 2"x") ("x"^2 + 1)^("x"^3 - 2"x" - 1) times "d"/"dx" ("x"^2 + 1) \$ \$ "f"'("x") = ("x"^3 - 2"x") ("x"^2 + 1)^("x"^3 - 2"x" -1) times ("d"/"dx" "x"^2 + "d"/"dx"1) \$ |
5 |
Step |
After differentiation |
\$ "f"'("x") = ("x"^3 - 2"x") ("x"^2 + "1")^("x"^3 - 2"x" - 1) times (2"x" + 0) \$ \$ "f"'("x") = ("x"^3 - 2"x") ("x"^2 + 1)^("x"^3 - 2"x" - 1) times (2"x") \$ |
6 |
Step |
Therefore, the derivative of f(x) is \$ "f"'("x") = ("x"^3 - 2"x") ("x"^2 + 1)^("x"^3 - 2"x" - 1) times (2"x")\$. |
|
7 |
Choice.A |
This option introduces the logarithmic term and subtracts instead of multiplying, making it incorrect |
\$(2"x"^4 - 3"x"^3 + 3"x"^2"ln"("x"^2 + 1) + "x"^2"ln"("x"^2 + 1) + 2"ln"("x"^2 + 1))(("x"^2 + 1)^("x"^3 - 2"x" - 1)) \$ |
8 |
Choice.B |
Option B arrives at the correct outcome despite not explicitly presenting the 2x factor. This is achieved through implicit differentiation within the composite term |
\$ "f"'("x") = ("x"^3 - 2"x") ("x"^2 + 1)^("x"^3 - 2"x" - 1) times (2"x")\$ |
9 |
Choice.C |
This option factors out 2x, but it misses a key multiplication within the composite term, leading to an incorrect application of the power rule |
\$ "f"'("x") = ("x"^3 - 2"x") ("x"^2 + 1)^("x"^3 - 2"x") times (2"x")\$. |
10 |
Choice.D |
This option shares the error of introducing \$ln(x^2 - 1)\$ and subtracting a term, making it incorrect |
\$ (2"x"^4 - 6"x"^2 - 3"x"^4"ln"("x"^2 + 1) - "x"^2"ln"("x"^2 + 1) + 2"ln"("x"^2 + 1))(("x"^2 + 1)^("x"^3 + 2"x" + 1)) \$ |
11 |
Answer |
Option |
B |
12 |
Sumup |
Can you summarize what you’ve understood in the above steps? |
|
Copyright © 2020-2022 saibook.us Contact: info@saibook.us Version: 1.5 Built: 19-April-2024 09:20AM EST