Question 33 of 70beginner🔧 ApplyNumerical1 mark

Find the derivative of 99x99x at x=100x = 100.

Correct Answer

99

Exercise: EXERCISE 12.2 | Q: 3 | (Page Page 32)
For More Understanding

Explanation

This question tests the understanding of derivative of a linear function. From the textbook context, Example 9 shows that the derivative of f(x) = 10x is found using the first principle: f'(x) = lim(h→0) [f(x+h) - f(x)]/h = lim(h→0) [10(x+h) - 10x]/h = lim(h→0) 10h/h = 10. Similarly, for f(x) = 99x, applying the same method gives f'(x) = 99. Since this derivative is a constant (independent of x), its value at x = 100 remains 99.

Solution Steps

  1. Step 1: Let f(x) = 99x. Using the first principle of derivative, f'(x) = lim(h→0) [f(x+h) - f(x)]/h

  2. Step 2: Substituting the function: f'(x) = lim(h→0) [99(x+h) - 99x]/h = lim(h→0) [99x + 99h - 99x]/h

  3. Step 3: Simplifying: f'(x) = lim(h→0) 99h/h = lim(h→0) 99 = 99

  4. Step 4: Since the derivative f'(x) = 99 is a constant, the derivative at x = 100 is 99.

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