Partial fractions then differentiate

Express a function in partial fractions, then use this form to find the derivative and prove a property (e.g., always negative, increasing/decreasing).

6 questions

Edexcel C34 2017 June Q5
5. $$\frac { 6 - 5 x - 4 x ^ { 2 } } { ( 2 - x ) ( 1 + 2 x ) } \equiv A + \frac { B } { ( 2 - x ) } + \frac { C } { ( 1 + 2 x ) }$$
  1. Find the values of the constants \(A , B\) and \(C\). $$f ( x ) = \frac { 6 - 5 x - 4 x ^ { 2 } } { ( 2 - x ) ( 1 + 2 x ) } \quad x > 2$$
  2. Using part (a), find \(\mathrm { f } ^ { \prime } ( x )\).
  3. Prove that \(\mathrm { f } ( x )\) is a decreasing function.
Edexcel P4 2021 October Q3
3. $$\mathrm { g } ( x ) = \frac { 3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 6 } { x ( x + 3 ) } \equiv A x + B + \frac { C } { x } + \frac { D } { x + 3 }$$
  1. Find the values of the constants \(A , B , C\) and \(D\). A curve has equation $$y = g ( x ) \quad x > 0$$ Using the answer to part (a),
  2. find \(\mathrm { g } ^ { \prime } ( x )\).
  3. Hence, explain why \(\mathrm { g } ^ { \prime } ( x ) > 3\) for all values of \(x\) in the domain of g .
OCR C4 2007 June Q1
1 The equation of a curve is \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 3 x + 1 } { ( x + 2 ) ( x - 3 ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence find \(\mathrm { f } ^ { \prime } ( x )\) and deduce that the gradient of the curve is negative at all points on the curve.
Edexcel PMT Mocks Q11
11. \(\frac { - 6 x ^ { 2 } + 24 x - 9 } { ( x - 2 ) ( 1 - 3 x ) } \equiv A + \frac { B } { x - 2 } + \frac { C } { 1 - 3 x }\)
a. Find the values of the constants \(A , B\) and \(C\).
b. Using part (a), find \(\mathrm { f } ^ { \prime } ( x )\).
c. Prove that \(\mathrm { f } ( x )\) is an increasing function.
Edexcel Paper 2 2018 June Q11
11. $$\frac { 1 + 11 x - 6 x ^ { 2 } } { ( x - 3 ) ( 1 - 2 x ) } \equiv A + \frac { B } { ( x - 3 ) } + \frac { C } { ( 1 - 2 x ) }$$
  1. Find the values of the constants \(A , B\) and \(C\). $$f ( x ) = \frac { 1 + 11 x - 6 x ^ { 2 } } { ( x - 3 ) ( 1 - 2 x ) } \quad x > 3$$
  2. Prove that \(\mathrm { f } ( x )\) is a decreasing function.
Edexcel C4 Q1
  1. The function \(f\) is given by
$$f ( x ) = \frac { 3 ( x + 1 ) } { ( x + 2 ) ( x - 1 ) } , x \in \mathbb { R } , x \neq - 2 , x \neq 1$$
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence, or otherwise, prove that \(\mathrm { f } ^ { \prime } ( x ) < 0\) for all values of \(x\) in the domain.