Question 24 of 71beginner🔧 ApplyNumerical1 mark

Find the derivative of xx at x=1x = 1.

Correct Answer

1

Exercise: EXERCISE 12.2 | Q: 2 | (Chapter: Page 32)
For More Understanding

Explanation

This question asks for the derivative of the function f(x) = x at x = 1. Following the definition of derivative from the textbook (Definition 1), we apply the first principle method. The derivative f'(a) is defined as the limit of [f(a+h) - f(a)]/h as h approaches 0. For the function f(x) = x, substituting values and simplifying yields f'(1) = 1. This matches the pattern seen in Example 5 of the context, where the derivative of 3x at x = 2 was found to be 3.

Solution Steps

  1. Step 1: Let f(x) = x. We need to find f'(1) using the definition: f'(1) = lim(h→0) [f(1+h) - f(1)]/h

  2. Step 2: Substitute f(1+h) = 1+h and f(1) = 1 into the formula: f'(1) = lim(h→0) [(1+h) - 1]/h

  3. Step 3: Simplify the numerator: f'(1) = lim(h→0) h/h = lim(h→0) 1

  4. Step 4: Since the limit of a constant is the constant itself: f'(1) = 1

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