Math Problem Statement
Give me the answer of this question with explanation
Solution
Let’s solve these derivatives step by step. These involve applications of the chain rule.
(i)
We use the chain rule here. Let , then .
-
Derivative of is:
-
Derivative of is:
Using the chain rule:
Final result:
(ii)
Again, applying the chain rule. Let , then .
-
Derivative of is:
-
Derivative of is:
Using the chain rule:
Final result:
(iii)
We proceed similarly. Let , then .
-
Derivative of is:
-
Derivative of is:
Using the chain rule:
Final result:
Final Answers:
Would you like any part of this explained in more detail?
Related Questions:
- What is the chain rule, and how is it applied in calculus?
- How do you differentiate in general?
- What happens if is a fractional power, like in part (iii)?
- Can the product rule and chain rule ever be applied together?
- How would you compute second derivatives of these functions?
Tip:
Always simplify inner derivatives first before multiplying them with the outer derivative—it avoids mistakes in complex chain rule problems.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Chain Rule
Formulas
Derivative of a power function: d/dx[u^n] = n*u^(n-1) * du/dx
Chain rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)
Theorems
Chain Rule of Differentiation
Suitable Grade Level
Grades 11-12