Question 49 of 70intermediate🔧 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): px2+qx+rax+b\frac{px^2 + qx + r}{ax + b}

Correct Answer

Let u=px2+qx+ru = px^2 + qx + r and v=ax+bv = ax + b.

Then u′=2px+qu' = 2px + q and v′=av' = a.

Using the quotient rule:

ddx(px2+qx+rax+b)=(2px+q)(ax+b)−(px2+qx+r)(a)(ax+b)2\frac{d}{dx}\left(\frac{px^2 + qx + r}{ax + b}\right) = \frac{(2px + q)(ax + b) - (px^2 + qx + r)(a)}{(ax + b)^2}

=2apx2+2bpx+aqx+bq−apx2−aqx−ar(ax+b)2= \frac{2apx^2 + 2bpx + aqx + bq - apx^2 - aqx - ar}{(ax + b)^2}

=apx2+2bpx+bq−ar(ax+b)2= \frac{apx^2 + 2bpx + bq - ar}{(ax + b)^2}

Exercise: Miscellaneous Exercise on Chapter 12 | Q: 9 | (Page Page 37)
For More Understanding

Explanation

This question is from the Miscellaneous Exercise on Chapter 12 (Limits and Derivatives). The question asks to find the derivative of a rational function of the form (px² + qx + r)/(ax + b). The solution requires applying the quotient rule, which states that for f(x) = u/v, the derivative is (u'v - uv')/v². The numerator is a quadratic polynomial and the denominator is a linear polynomial. After differentiating both and substituting into the quotient rule formula, we simplify by expanding and combining like terms.

Solution Steps

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

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

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

  4. Step 4: Expand numerator: (2px + q)(ax + b) - a(px² + qx + r)

  5. Step 5: Simplify to get final answer: (apx² + 2bpx + bq - ar)/(ax + b)²

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