GATE CS 2016 Set 2 — Question 12
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Engineering Mathematics → Calculus → Differentiation
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Question
Let be a polynomial and be its derivative. If the degree of is 10, then the degree of is ________.
Correct answer
9 to 9
Solution
Let . The expression isolates the even powers of in (since odd powers cancel out). If the degree of is 10, then the highest even power in is .Let . The expression isolates the odd powers of in (since even powers cancel out).
The terms in come from differentiating terms in .
The highest even power in is . Its derivative is , which is an odd power of degree 9.
Any higher powers in would have to be odd (e.g., ), and their derivatives would be even (e.g., ), which would cancel out in .Thus, the degree of is 9.
The terms in come from differentiating terms in .
- The derivative of an even power is (an odd power).
- The derivative of an odd power is (an even power).
The highest even power in is . Its derivative is , which is an odd power of degree 9.
Any higher powers in would have to be odd (e.g., ), and their derivatives would be even (e.g., ), which would cancel out in .Thus, the degree of is 9.
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