Exponential or trigonometric base functions

Questions where the base function is exponential, trigonometric, or their products (e.g., e^x sin x, e^(cos²x), tan²x), requiring standard differentiation rules to establish the derivative relationship.

5 questions

CAIE Further Paper 2 2020 June Q2
2 It is given that \(y = 2 ^ { x }\).
  1. By differentiating \(\ln y\) with respect to \(x\), show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = 2 ^ { \mathrm { x } } \ln 2\).
  2. Write down \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\).
  3. Hence find the first three terms in the Maclaurin's series for \(2 ^ { X }\).
Edexcel F2 2017 June Q5
5. $$y = \mathrm { e } ^ { \cos ^ { 2 } x }$$
  1. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { \cos ^ { 2 } x } \left( \sin ^ { 2 } 2 x - 2 \cos 2 x \right)$$
  2. Hence find the Maclaurin series expansion of \(\mathrm { e } ^ { \cos ^ { 2 } x }\) up to and including the term in \(x ^ { 2 }\)
Edexcel F2 2024 June Q7
  1. Given that \(y = \mathrm { e } ^ { x } \sin x\)
    1. show that
    $$\frac { \mathrm { d } ^ { 6 } y } { \mathrm {~d} x ^ { 6 } } = k \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$$ where \(k\) is a constant to be determined.
  2. Hence determine the first 5 non-zero terms in the Maclaurin series expansion for \(y\), giving each coefficient in simplest form.
OCR FP2 2013 January Q5
5 You are given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { - x } \sin x\).
  1. Find \(f ( 0 )\) and \(f ^ { \prime } ( 0 )\).
  2. Show that \(\mathrm { f } ^ { \prime \prime } ( x ) = - 2 \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x )\) and hence, or otherwise, find \(\mathrm { f } ^ { \prime \prime } ( 0 )\).
  3. Find a similar expression for \(\mathrm { f } ^ { \prime \prime \prime } ( x )\) and hence, or otherwise, find \(\mathrm { f } ^ { \prime \prime \prime } ( 0 )\).
  4. Find the Maclaurin series for \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 3 }\).
WJEC Further Unit 4 Specimen Q12
12. The function \(f\) is given by $$f ( x ) = \mathrm { e } ^ { x } \cos x$$
  1. Show that \(f ^ { \prime \prime } ( x ) = - 2 \mathrm { e } ^ { x } \sin x\).
  2. Determine the Maclaurin series for \(f ( x )\) as far as the \(x ^ { 4 }\) term.
  3. Hence, by differentiating your series, determine the Maclaurin series for \(\mathrm { e } ^ { x } \sin x\) as far as the \(x ^ { 3 }\) term.
  4. The equation $$10 \mathrm { e } ^ { x } \sin x - 11 x = 0$$ has a small positive root. Determine its approximate value, giving your answer correct to three decimal places.