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234 Math Practice Problems: Calculate Derivatives 1. Calculate
d (f (x)) dx
(also known as f (x)) of these functions
a. f (x) = 3 b. f (x) = x2 c. f (x) = x + 1 d. f (x) = 2x3 + 5x2 4x 1 e. f (x) = 10x4 3x3 + 2x2 5x + 10 f. f (x) = x 2 g. f (x) = x h. f (x) = 3 x i. f (x) = 5 4 x + 4x2 j. f (x) = x1 k. f (x) = l. f (x) =
1 x 1 x2 2 x
1
m. f (x) = 5x3 2x + 55 n. f (x) = x o. f (x) = 3x p. f (x) = x3 q. f (x) = 4x3
+
3 x4
r. f (x) = x4 Note that we have a pattern here, where given any constant d d c R, dx (cf (x)) = c dx (f (x)). You should make use of this fact to simplify future calculations - it can be the dierence from blindly using product or quotient rule and simply applying the power rule for some!
1
2
s. f (x) = x(x + 1) t. f (x) = 2x(x + 1) u. f (x) = (x + 1)(x 1) v. f (x) = (x + 2)(x + 5) w. f (x) = 3(x 3)(x 4) x. f (x) = 4(x2 + 10x)(x 15) y. f (x) = (x + 1)(x + 2)(x + 3) z. f (x) = (x 4)(x 5)(x 6) . f (x) = x2 (x + 2) . f (x) = x3 (x 1) . f (x) = x(2x 1) . f (x) = 2x2 (3x2 + 2x + 1) . f (x) = (4x3 + 3x2 + 2x + 1)(x2 x 1) . f (x) = (x4 + x3 + x2 + x + 1)2 (x 1)2 . f (x) = (x + 1)3 (x 1)3 . f (x) = (2x 1)(3x + 1)3 (10x 1)2 . f (x) = (x2 + 1)2 (2x3 + x2 1)3 (3x4 + + 3x2 2x + 1)4 . f (x) = . f (x) = . f (x) = . f (x) = . f (x) =
x+1 x x x2 2 (x+1)2 (x1)3 (x3 +3x1) (5x2 +x+1)2 (3x2 +1)2 (2x+1) x2
. f (x) =
3
3 +x x 3 2 +9 x
. f (x) =
. f (x) = ( x2x )2 (10x2 + 10 + 1)3 1
x . f (x) = ( x+1 + x3 )4 (4x3 2x2 4)
2
. f (x) =
x (x x+1
+ 2)
. f (x) = (4x2 + 1) x1 4x . f (x) = 5x2 + 3x . f (x) = . f (x) =
5
10x1 x2 +2
(3x1)(5x5 3x)2 r 2 10x2 3 + 3x 3 2
x 3x 1
*
(5x+1)4
* denotes a challenge problem that tests your knowledge on usage of all of the rules . f (x) = ex . f (x) = ex
3 +x
. f (x) = ex + ex . f (x) = (ex + 1)(ex 1) . f (x) = e3x (2x + 1)2 . f (x) = . f (x) = . f (x) =
e5x +1 10x3 (3x+2)3 e5x3
2 2 2 2
e3x
5 +x
. f (x) = (e6x . f (x) = ln x
)2
. f (x) = ln(10x + 1) 1. f (x) = ln(14x5 + 3x)
4
2. f (x) = ln ex 3. f (x) = ln e5x 4. f (x) = (ln(x2 + 1) + 1)(ln(x) + 1) 5. f (x) = (ln(3x 1))2 (6x2 + 1)2 6. f (x) = 7. f (x) = 8. f (x) =
ln 5x 4x ln(x2 1) ln(x2 +2x+1)
ln(x3 1)
1
9. f (x) = (ln(x2 + 3x)) 3 10. f (x) = e( ln x) 11. f (x) = eln(x
2 +1)
12. f (x) = (ln(ex ) 1)(eln(x ) + 1) 13. f (x) = ex
3 +x2
2
2
ln(2x + 1) ln(4x + 3) + x2 (3x + 4)
14. f (x) = e3x 15. f (x) =
4 +4x3
6x+1 x1
e2x+3 ln(x2 1)
16. (ex + ex ) ln(4x2 1)

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