Math Problem Statement
etermine the following derivatives: (i) d dx (∫ x 2 t sin t dt )
Solution
To find the derivative of the given expression, let's first rewrite it clearly:
To solve this, we can apply the Leibniz rule for differentiation under the integral sign, which is a special case of the Fundamental Theorem of Calculus. The Leibniz rule states:
Here, the integral is from a constant to . Therefore, the derivative simplifies as:
Since and , the derivative is:
Thus, the derivative is:
Would you like further details or have any questions about the solution?
Here are some related questions you might consider:
- What is the derivative of ?
- How does the Leibniz rule apply when the limits of integration are both functions of ?
- Can you compute the derivative ?
- What if the integrand involved explicitly, like ?
- How would you differentiate when both limits are functions of ?
Tip: When applying the Leibniz rule, always identify if the limits of integration are constants or functions of , as this affects how the rule is applied.
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Math Problem Analysis
Mathematical Concepts
Integral Calculus
Differentiation
Leibniz Rule
Formulas
Leibniz rule for differentiation under the integral sign
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
College Level
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