Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Functions - Combined functions.
Calculus - Differentiation - Test Yourself 1.


 

Before you start the serious stuff, read the following to show how logs and exponentials can combine to make you happy!!!!

Now to the serious stuff:

Exponential and log functions. 1. Differentiate and simplify. 2. Differentiate

.
  3. Differentiate . 4. (i) Show that

xex = ex+ln x

(ii) Hence differentiate xex without using the product rule.
  5. Differentiate . 6. Differentiate .
 

7. Differentiate

.

8. Find the coordinates of the point on the curve y = e3x where the tangent is perpendicular to the line

x + 6y - 24 = 0.
Exponential and trig functions. 9. Differentiate . 10. Differentiate .
  11. Show that if y = sinx etanx

then the derivative is

y' = etanx(cosx + tanx secx).

 12. Differentiate .
  13. Differentiate y = sin (e2x). 14. Differentiate y = 2e tan 3x
  15. For the function f(x) = e-x cos x defined in the domain [0, 2π]:

(i) Find the x values for the stationary points and determine their nature.

(ii) Sketch the curve showing the coordinates on the stationary points (y values to 3 decimal places) and the intercepts with the two axes.

(iii) Find the number of solutions to the equation
e-x cos x - 0.5x = 0 in the given domain. Explain your answer.

Answer.(i) SPs at x = 3π/4 and
at x = 7π/4.
(ii) Min SP at x = 3π/4 and
max SP at x = 7π/4.
(iv) There is only one solution -
see the graph.
Log and trig functions. 16. Show that

.

 17. .
  18. Show that the derivative of

is

19. Show that the derivative of

can be expressed as

 

20. Show that the derivative of
y = tan (ln x2) is .

 21. Differentiate .
 

22. Find the derivative of

 23.