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# Derivatives of Products and Quotients

Lesson Plans > Mathematics > Calculus > Differential

## Derivatives of Products and Quotients

In the calculus curriculum I'm teaching, we arrive at derivatives of products and derivatives of quotients before we learn the derivatives of such interesting functions as the trig functions and the exponential and log functions. This is, in some ways, unfortunate, as these functions will give us more interesting practice problems in taking derivatives. For example, if we know the derivative of sin and the derivative of cos, students can use the quotient rule to find the derivative of every other trig function. Because of this, I developed the worksheet below, which provides the student with four basic derivatives that they may not already know, and invites them to use that information in conjunction with the product and quotient rule to calculate other derivatives. The worksheet is generously sprinkled with hints on how students might want to approach the problem.

The derivatives included in the worksheet are shown below:

d
dx
(sin x) = cos x
d
dx
(cos x) = -sin x
d
dx
(ex) = ex
d
dx
(ln x) =
1
x

After doing this worksheet, it's a worthwhile activity to ask students if, based on questions 9 and 10, they can make a prediction of what the derivative of ekx would be, for any constant k. The obvious prediction can be easily proved for all integers k, using an inductive proof.

A similar project could be undertaken for the function lnk(x), or for sink(x).

Lesson by Mr. Twitchell

## Products and Quotients Worksheet

Find the derivatives of the following products and quotients, given the following derivatives:
d
dx
(sin x) = cos x;
d
dx
(cos x) = -sin x;
d
dx
(ex) = ex
d
dx
(ln x) =
1
x

1. xsin x

2. (cos x)(ln x)

3. (2x + 1)(x - 5)

4. 2x + 1
x - 5

5. sin x
ex

6. csc x [Hint: write in terms of other trig functions]

7. sec x

8. cot x

9. e2x [Hint: write this as (ex)(ex)]

10. e3x [Hint: write this as (ex)(e2x)]

11. sin2x [Hint: write this as (sin x)(sin x)]

12. x(ln x)

13. (ln x)2

14. sin 2x [Hint: use the double-angle formula]

15. cos 2x

16. (sin x)(cos x)ex [Hint: choose two functions to work with, then incorporate the third, once you've found the derivative)

17. (sin x)(cos x)
ln x

## Products and Quotients Worksheet: Answer Key

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