Question 51 of 70beginner🔧 ApplyShort Answer1 mark

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 4x−24\sqrt{x} - 2

Correct Answer

Let f(x) = 4√x - 2 = 4x1/24x^{1/2} - 2

Differentiating with respect to x: f'(x) = 4 × (1/2)x1/2−1x^{1/2 - 1} - 0 f'(x) = 2x−1/22x^{-1/2} f'(x) = 2/√x

Therefore, the derivative is 2/√x.

Exercise: Miscellaneous Exercise on Chapter 12 | Q: 11 | (Page Page 37)
For More Understanding

Explanation

This is a differentiation problem from the list of functions in the textbook context. The function 4√x - 2 needs to be differentiated using the power rule. Writing √x as x1/2x^{1/2}, we apply the derivative formula d/dx(xnx^{n}) = nxn−1nx^{n-1}. The constant term -2 differentiates to 0. The context shows similar applications of the power rule in limit evaluations.

Solution Steps

  1. Step 1: Rewrite the function as f(x) = 4x1/24x^{1/2} - 2

  2. Step 2: Apply power rule: d/dx(xnx^{n}) = nxn−1nx^{n-1}

  3. Step 3: Differentiate: f'(x) = 4 × (1/2)x−1/2x^{-1/2} - 0 = 2x−1/22x^{-1/2}

  4. Step 4: Simplify: f'(x) = 2/√x

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