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

Exponential functions - differentiation.
Test Yourself 1.


 

The questions on this page focus on:
1. Differentiating basic exponential functions.
2. Differentiating exponentials with the chain rule.
3. Differentiating exponentials with the product rule.
4. Differentiating exponentials with the quotient rule.
5. Differentiating exponentials to find tangents.
6. Differentiating exponentials to find normals.
7. Miscellaneous differentiation of exponentials.

 

Differentiate each of the following equations unless there is another instruction:

(follow this hyperlink for differential equations with exponentials).

Basic expressions 1. y = 2ex
Answer.2ex.		 2. y = 3e5x + 42
Answer.15e5x.
  3.
Answer.0.25ex/4.		 4. ex - e-2x
Answer.ex + 2ex.
  5. y = e5
Answer.0.		 6.
Answer.-12/e6x.
  7. 8.
Chain rule 9. y = (1 + 2ex)4
Answer.8ex(1 + 2ex)3.		 10. y = (1 - 3e5x)3
Answer.-90e5x(1 - 3e5x)2.
  11. y = 4(3 + e2x)5
Answer.40e2x(3 - e2x)4.		 12. y = (e-x - ex)2
Answer.-2(e-x - ex)(e-x + ex).
Product rule. 13. y = x2 ex
Answer.Gradient = xex(2 + x).		 14. m = t2 - t3e2t
Answer.2t - t2e2t(3 + 2t).
  15. y = (2x + 1)e-x

Answer.Gradient = e-x(1 - 2x).

 16.
Quotient rule. 17. 18.
  19.
Answer.4/(ex + e-x)2.		 20.
Answer.(1 + 2x)/e2x.
Tangents 21. Determine the gradient of the tangent to the curve where x = 1.
Answer.Gradient = 4.		 22. Show that the tangent to the curve y = ex - 2x at the point
(1, e - 2) passes through the origin.
  23. Find the equation of the tangent to the curve y = e2x + x at the point with the x co-ordinate
of 0.

Answer.Tangent is 3x - y + 1 = 0.		 24. Find the equation of the tangent to y = e-2x at (0, 1).
Answer.Tangent is 2x + y - 1 = 0.
  25. For what values of x is the tangent to y = e3x parallel to the line y = 6x?
Answer.At x = (ln 2)/3.		 26. Find the equation of the tangent to the curve y = 2x ex at the
point where x = 1.
Answer.Tangent is 6ex - y - 4e = 0.
Normals 27. Find the equation of the normal to the curve y = e2x-1 at the point where x = 2 leaving your answer in exact form.
Answer.Normal is y = -x/2e3 + e3 + 1/e3.		 28. (i) Find the equation of the normal to at the point where x = 1.

(ii) Find where that normal crosses the x-axis (to 2 significant figures).

Answer.x = -0.083.
Miscellaneous. 29. Find the x values for which the curve y = x2 e-x is decreasing.
Answer.x < 0 or x > 2.		 30. Find the values for x for which
the gradient of y = xe-0.5x is greater than 0.
Answer.x < 2.