Question 48 of 70intermediate🔧 ApplyShort Answer1 mark

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,ra, b, c, d, p, q, r and ss are fixed non-zero constants and mm and nn are integers): ax+bpx2+qx+r\frac{ax + b}{px^2 + qx + r}

Correct Answer

Let f(x) = ax+bpx2+qx+r\frac{ax + b}{px^2 + qx + r}

Using the quotient rule:

f'(x) = a(px2+qx+r)−(ax+b)(2px+q)(px2+qx+r)2\frac{a(px^2 + qx + r) - (ax + b)(2px + q)}{(px^2 + qx + r)^2}

= apx2+aqx+ar−2apx2−aqx−2bpx−bq(px2+qx+r)2\frac{apx^2 + aqx + ar - 2apx^2 - aqx - 2bpx - bq}{(px^2 + qx + r)^2}

= −apx2−2bpx+ar−bq(px2+qx+r)2\frac{-apx^2 - 2bpx + ar - bq}{(px^2 + qx + r)^2}

Exercise: Miscellaneous Exercise on Chapter 12 | Q: 8 | (Page Page 37)
For More Understanding

Explanation

This question requires applying the quotient rule for differentiation. The function is in the form u/v where u = ax + b and v = px² + qx + r. The quotient rule states that d/dx(u/v) = (u'v - uv')/v². Here u' = a and v' = 2px + q. After applying the rule and simplifying, we get the final derivative expression.

Solution Steps

  1. Step 1: Identify u = ax + b and v = px² + qx + r

  2. Step 2: Find derivatives: u' = a and v' = 2px + q

  3. Step 3: Apply quotient rule: f'(x) = (u'v - uv')/v²

  4. Step 4: Substitute and simplify the numerator

  5. Step 5: Write final answer with denominator squared

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