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Differentiation Worksheet

Differentiation Worksheet
• Page 1
1.
If $f$($u$) = $\frac{5{u}^{2}-9u}{9u+4}$, then $f$ ′($u$)
 a. b. $\frac{10u-9}{9}$ c. -41 + 135$u$2 d.

Solution:

f ′ (u) = (9u + 4)(10u - 9) - (5u2 - 9u)9(9u + 4)2
[Use Quotient rule.]

= 90u2 + 40u - 81u - 36 - 45u2 + 81u(9u + 4)2

= 45u2 + 40u - 36(9u + 4)2

2.
If $g$($t$) = $\frac{3{t}^{3}+4t-5}{4t-5}$, then $g$′(t)
 a. b. c. d.

Solution:

g′(t) = (4t - 5)(9t2 + 4) - (3t3 + 4t - 5)(4)(4t - 5)2
[Use the quotient rule and the chain rule.]

= 36t3 - 45t² + 16t - 20 - 12t3 - 16t + 20(4t - 5)2
[Simplify.]

= 24t3 - 45t2(4t - 5)2
[Simplify.]

3.
If $y$ = 6$x$${e}^{2x}$, then $y$′ = ?
 a. 6$x$$e$2$x$ + 6$e$2$x$ b. 36$x$ + 12$e$4$x$ c. 12$x$2$e$2$x$ - 1 + 6$e$2$x$ d. 6 ${e}^{2x}$ (2$x$ + 1)

Solution:

y = 6xe2x

y′ = 6x(2) e2x + e2x (6)
[Use the product rule.]

= 12x e2x + 6e2x

= 6e2x (2x + 1)

4.
If $y$ = - 8 , then find $y$′.
 a. 16$x$${e}^{-2x}\left(1+4x\right)$ b. 8$x$ (2$x$ - 2) c. 8 d. - 8${e}^{-8x}\left({x}^{2}+2x\right)$

Solution:

y′ = - 8x2.(- 2e- 2x) + e- 2x (- 16x)
[Use product rule.]

= 16x2e- 2x - 16x e- 2x

= 8x e- 2x (2x - 2)

5.
If $y$ = $\frac{\mathrm{ln}\left(2x\right)}{4{x}^{2}-4}$, then find $y$′.
 a. b. $\frac{{x}^{2}\left[1-2\mathrm{ln}\left(2x\right]\right)-4}{x}$ c. d.

Solution:

y′ = (4x2 - 4)[22x]-(ln 2x)(8x)(4x2-4)2
[Use the quotient rule.]

= 4x -4x - 8x(ln 2x)(4x2 -4)2

= 4x2[1-2ln(2x)]-4x(4x2 - 4)2
[Multiply numerator and denominator with x and simplify.]

6.
If $g$($x$) = , then what is $g$′ ($x$)?
 a. b. c. d.

Solution:

g′(x) = (6x + 5)(- 10x) - (- 5x2)(6)(6x + 5)2
[Use the quotient rule.]

= - 60x2 - 50x + 30x2(6x + 5)2

= - 30x2 - 50x(6x + 5)2

7.
If $y$ = , then find $y$′.
 a. b. c. d. none of the above

Solution:

y′ = (x + 6)[12x-12 ]-(x12 - 3)(x + 6)2
[Use the quotient rule.]

= [12x12 +62x-12 -x12 + 3(x + 6)2] [2 / 2]
[Multiply and divide by 2.]

= x12 + 6x-12 - 2x12 + 62(x + 6)2 x12x12
[ Multiply and divide by x12.]

= - x + 6 + 6x122x12(x + 6)2

= 6 - x + 6x122x12(x + 6)2

= 6 - x + 6x2x(x + 6)2

8.
If $y$ = , then $y$′ = ?
 a. 3$x$2 - 8$x$ + 6 b. c. 2$x$ - 2 d.

Solution:

y′ = (x - 2)(2x - 2) - (x2 - 2x + 2)(1)(x - 2)2

= x2- 4x + (4 - 2)(x - 2)2

= x2 - 4x + 2(x - 2)2

9.
If $y$ = $\frac{7{x}^{2}+2x-6}{x-7}$, then $y$′ = ?
 a. $\frac{{x}^{2}-98x-8}{{\left(x-7\right)}^{2}}$ b. $\frac{7{x}^{2}-98x-8}{{\left(x-7\right)}^{2}}$ c. $\frac{7{x}^{2}-8}{{\left(x-7\right)}^{2}}$ d. $\frac{7{x}^{2}-98x}{{\left(x-7\right)}^{2}}$

Solution:

y′ = (x-7) (14x+2)-(7x2+2x-6)(1)(x-7)2
[Use the quotient rule.]

= 14x2+2x-98x-14-7x2-2x+6(x-7)2

= 7x2-98x-8(x-7)2
[Simplify.]

10.
If $y$ = , then $y$′ = ?
 a. $\frac{18{x}^{2}-162x+14}{{\left(x-9\right)}^{2}}$ b. c. $\frac{18{x}^{2}}{\left(x+9\right)}$ d.

Solution:

y′ = (x+9)(18x-2)-(9x2-2x-4)(1)(x+9)2
[Use quotient rule.]

= 18x2+162x-2x-18-9x2+2x+4(x+9)2

= 9x2+162x-14(x+9)2
[Simplify.]