Question 46 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+bcx+d\frac{ax + b}{cx + d}

Correct Answer

Let f(x) = ax+bcx+d\frac{ax + b}{cx + d}. Using the quotient rule, we have:

f'(x) = (ax+b)′(cx+d)−(ax+b)(cx+d)′(cx+d)2\frac{(ax + b)'(cx + d) - (ax + b)(cx + d)'}{(cx + d)^2}

= a(cx+d)−c(ax+b)(cx+d)2\frac{a(cx + d) - c(ax + b)}{(cx + d)^2}

= acx+ad−acx−bc(cx+d)2\frac{acx + ad - acx - bc}{(cx + d)^2}

= ad−bc(cx+d)2\frac{ad - bc}{(cx + d)^2}

Exercise: Miscellaneous Exercise on Chapter 12 | Q: 5 | (Page Page 37)
For More Understanding

Explanation

This question from the Miscellaneous Exercise on Chapter 12 requires applying the quotient rule for differentiation. The context shows similar applications in Example 22 where quotients are differentiated. The numerator derivative is 'a' (derivative of ax + b) and denominator derivative is 'c' (derivative of cx + d). After applying the quotient rule formula, the terms with 'acx' cancel out, leaving the constant expression (ad - bc) in the numerator.

Solution Steps

  1. Step 1: Identify u(x) = ax + b and v(x) = cx + d

  2. Step 2: Find u'(x) = a and v'(x) = c

  3. Step 3: Apply quotient rule: f'(x) = u′v−uv′v2\frac{u'v - uv'}{v^2}

  4. Step 4: Simplify: a(cx+d)−c(ax+b)(cx+d)2\frac{a(cx + d) - c(ax + b)}{(cx + d)^2}

  5. Step 5: Expand and cancel terms to get ad−bc(cx+d)2\frac{ad - bc}{(cx + d)^2}

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