Question 48 of 71intermediate🔧 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): (px+q)(rx+s)(px + q)\left(\frac{r}{x} + s\right)

Correct Answer

ps - qrx2\frac{qr}{x^2}

Exercise: Miscellaneous Exercise on Chapter 12 | Q: 3 | (Chapter: Page 37)
For More Understanding

Explanation

We need to differentiate the product (px+q) and (r/x + s). Using the product rule, derivative = (px+q)'(r/x + s) + (px+q)(r/x + s)'. The derivative of (px+q) is p. The derivative of (r/x + s) is -r/x2x^{2} (since derivative of 1/x is -1/x2x^{2} and s is constant). Substituting and simplifying yields ps - qr/x2x^{2}.

Solution Steps

  1. Step 1: Let f(x) = (px+q) and g(x) = (r/x + s). Then by product rule, d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x).

  2. Step 2: Compute f'(x) = d/dx (px+q) = p.

  3. Step 3: Compute g'(x) = d/dx (r/x + s) = d/dx (r x−1x^{-1}) + 0 = r * (-1) x−2x^{-2} = -r/x2x^{2}.

  4. Step 4: Apply product rule: d/dx = p * (r/x + s) + (px+q) * (-r/x2x^{2}).

  5. Step 5: Expand: = p r/x + p s - (px+q) r / x2x^{2} = p r/x + p s - p r/x - q r/x2x^{2}.

  6. Step 6: Cancel p r/x terms to obtain the final derivative: p s - q r/x2x^{2}.

PreviousNext