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Calculus - Differentiation - Basic techniques.
Test Yourself 1.

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Find the gradient function of the following functions - that is "differentiate each of the following functions" with respect to the given variable. All questions use only the direct technique as described - no special rules.

As always - your first step is to look at the structure and decide what you need to do.

The questions below focus on the structures of:

1. the basic format.

2. the use of simple brackets.

3. the use of the d/dx format.

4. radicals.

5. terms in the denominator.

6. fractions with only one term in the denominator.

1. Basic format.	1. y = x3 + x Answer.3x2 + 1.	2. y = 3x3 + 2x2 - x - 42 Answer.9x2 + 4x - 1.	3. y = 4x2 + 5x Answer.3x2 + 1.
	4. y = 3t - 5t2 Answer.3 - 10t.	5. y = 0.5x4 + 1.5x2 - 42 Answer.2x3 + 3x.	6. y = x2.5 Answer.2.5x1.5.
2. Simple brackets.	7. s = 2t2(3t + 4) Answer.18t2 + 16t	8. m = 4n3(n5 + 3n - 1) Answer.32n7 + 48n3 - 12n 2	9. y = 3x3(2x4 - 5x2 - x) Answer.42x6 - 75x4 - 12x3
3. Use of d/dx format.	10. Answer.3t2 + 10t - 7	11. Answer.3u2 + 10u - 12	12. Answer.-9z-4
	13. Answer.-2 - 12x2	14. Answer.y - 5	15. Answer.v3 - 4v The number 4 is a decoration - leave it where it is. Rewrite the square root as an exponent and combine with the other x term in the denominator.Move the combined x term to the numerator using a negative exponent. Then differentiate normally.
4. Radicals.	16. Hint.Rewrite the square root as an exponent (1/2)and add to the index of 2 in the first term to give x5/2. Then differentiate normally. Answer.2.5x1.5	17. Answer.-2.5 x -3.5	18. Hint.Rewrite the fifth root as an exponent and use a negative index to bring the t term into the numerator. Combine the exponents. Then differentiate normally.
(see the Solutions for the answers to Q 18 - 21).	19.	20.	21.
5. terms in the denominator - so the neeed to use negative signs.	22. Hint.The 5 is a number in the denominator - leave it there. Use a negative index to bring the x term into the numerator. Then differentiate normally. Answer.-2 /(5x 3)	23. Answer.-3 /(4x2)	24. Hint.Leave the 4 and the 3 where they are. Combine the x terms to give x3/2 and bring that to the numerator with a negative index. Differentiate normally.
	25. Answer.-2 /(x + 5)2	26. Answer.-8 /(2t - 7)2	27. Answer.-18 /(1 - 2t)4
6. Fractions with 1 term in the denominator.	28. Hint.How many terms in the denominator? ONE. So divide each term in the numerator by the term in the denominator and then differentiate normally. Answer.4 - 4x-3	29. Answer.1 + 3x -2 - 6x -4	30. Answer.9/(5x 8) -2x3 + 3 /(10x2)