﻿ Derivatives Using Definition Worksheet | Problems & Solutions

# Derivatives Using Definition Worksheet

Derivatives Using Definition Worksheet
• Page 1
1.
Find the derivative of the function ($y$) = 9$y$ + 14 directly from the definition of derivative.
 a. 14 b. 9 c. $\frac{1}{9}$ d. - 9

#### Solution:

(y) = 9y + 14
[Write the function.]

′ (y) = limh0 f (y+h)-f (y)h
[Use the definition of derivative.]

= limh0 (9(y+h)+14)-(9y+14)h
[Use the definition of (y).]

= limh0 (9hh) = 9
[Evaluate the limit.]

′ (y) = 9

2.
Use the definition of the derivative to find the derivative of $g$($s$) = 4$s$2 + 6$s$ + 46.
 a. 8$s$ b. 8 c. 4$s$ + 6 d. 8$s$ + 6 e. 4$s$

#### Solution:

g(s) = 4s2 + 6s + 46
[Write the function.]

g′ (s) = limh0 g(s+h)-g(s)h
[Use the definition of derivative.]

= limh0 (4(s+h)2+6(s+h)+46)-(4s2+6s+46)h
[Use the definition of g(s).]

= limh0 (4h2+8sh+6hh) = 8s + 6
[Evaluate the limit.]

g′ (s) = 8s + 6

3.
From the definition of derivative, find the derivative of ($v$) = 6$v$5/9 + 26
 a. $\frac{5}{9}$$v$ - 4/9 b. $v$ - 4/9 c. - $\frac{10}{3}$$v$ - 4/9 d. $\frac{54}{5}$$v$ 4/9 e. $\frac{10}{3}$$v$ - 4/9

#### Solution:

(v) = 6v5/9 + 26
[Write the function.]

′ (v) = limh0f (v+h)-f (v)h
[Use the definition of of ′ (v).]

= limh0(6(v+h)5/9+26)-(6v5/9+26)h
[Use the definition of (v).]

= 6(limh0(v+h)5/9 -v5/9h)

= 10 / 3v - 4/9
[Evaluate the limit.]

′ (v) = 10 / 3v - 4/9

4.
Use the definition of the derivative to find $g$′ ($x$) if $g$($x$) = 11$x$9 + 15.
 a. 9$x$8 b. 99$x$8 c. 99$x$9 d. 11$x$8 e. 99$x$8 + 15

#### Solution:

g(x) = 11x9 + 15
[Write the function.]

g′ (x) = limh0 g(x+h)-g(x)h
[Use the definition of the derivative.]

= limh0 (11(x+h)9+15)-(11x9+15)h
[Use the definition of g(x).]

= 11(limh0 (x+h)9-x9h)

= 11(9x8) = 99x8
[Evaluate the limit.]

g′ (x) = 99x8

5.
Find the derivative of ($t$) = $\frac{18}{\sqrt{6t+27}}$ using the definition of derivative.
 a. - 9 (6$t$ + 27)- 3/2 b. - 54 (6$t$ + 27)- 3/2 c. - $\frac{1}{9}$(6$t$ + 27)- 3/2 d. (6$t$ + 27)- 3/2 e. - (6$t$ + 27)- 3/2

#### Solution:

(t) = 186t+27
[Write the function.]

′ (t) = limh0 f (t+h)-f (t)h
[Use the definition of ′ (t).]

= limh0 (186(t+h)+27-186t+27h)
[Use the definition of (t).]

= 18 limh0 (6t+27-6(t+h)+27h (6(t+h)+27) (6t+27))

= - 54 (6t + 27)- 3/2
[Evaluate the limit.]

′ (t) = - 54 (6t + 27)- 3/2

6.
Find the derivative of $g$($v$) = 11$\sqrt{v}$ + 13.
 a. $\frac{11\sqrt{v}}{2}$ b. $\frac{1}{2\sqrt{v}}$ c. $\frac{11}{\sqrt{v}}$ d. - $\frac{11}{2\sqrt{v}}$ e. $\frac{11}{2\sqrt{v}}$

#### Solution:

g(v) = 11v + 13
[Write the function.]

g ′ (v) = limh0 g(v+h)-g(v)h
[Use the definition of g′ (v).]

= limh0 (11v+h+13)-(11v+13)h
[Use the definition of g(v).]

= 11(limh0 v+h-vh)

= 11 limh0(1v+h+v) = 112v
[Rationalize the numerator and evaluate the limit.]

g ′ (v) = 112v

7.
Find the derivative of $g$($v$) = 7tan 6$v$ from the definition of derivative.
 a. 42 sec 6$v$ b. sec2 6$v$ c. 42 cosec2 6$v$ d. 42 sec2 6$v$ e. $\frac{1}{42}$cos2 6$v$

#### Solution:

g(v) = 7tan 6v
[Write the function.]

g′(v) = limh0 g(v+h)-g(v)h
[Use the definition of g′ (v).]

= limh0 (7tan 6(v+h)-7tan 6vh)
[Use the definition of g(v).]

= 7 limh0 (sin 6hhcos 6(v+h)cos 6v)
[Use tan A - tan B = sin (A-B)cos A cos B]

= 7 (limh0 (sin 6hh)) (limh0 1cos 6(v+h)cos 6v)
[Use the Product Rule of limits.]

= (7) (6) 1(cos 6v)2 = 42 sec2 6v
[Use limx0 sin kxx = k and evaluate the limits.]

g′ (v) = 42 sec2 6v

8.
Find ′ ($p$), if ($p$) = 8cot 7$p$ + 12 from the definition of derivative.
 a. $\frac{1}{56}$sin2 7$p$ b. - 56 cosec2 7$p$ c. 56 cosec2 7$p$ d. - 7 cosec2 7$p$ e. - 8 cosec2 7$p$

#### Solution:

(p) = 8cot 7p + 12
[Write the function.]

′(p) = limh0 f (p+h)-f (p)h
[Use the definition of ′ (p).]

′ (p) = limh0 (8cot 7(p+h)+12)-(8cot 7p+12)h
[Use the definition of (p).]

= 8 limh0 cot 7(p+h)-cot 7ph

= 8 limh0 (sin (- 7h)h sin 7(p+h)sin 7p)
[Use cot A - cot B = sin (B-A)sin A sin B]

= (- 8) (limh0 sin 7hh) (limh0 1sin 7(p+h) sin 7p)
[Use the Product Rule of limits.]

= (- 8) (7) (1(sin 7p)2) = - 56 cosec2 7p
[Use limx0 (sin kxx) = k and evaluate the limits.]

′ (p) = - 56 cosec2 7p

9.
Find the derivative of ($x$) = using the definition of derivative.
 a. $\frac{144}{{\left(9x+12\right)}^{2}}$ b. c. d. $\frac{16}{9}$${\left(9x+12\right)}^{2}$ e. $\frac{9}{{\left(9x+12\right)}^{2}}$

#### Solution:

(x) = 169x+12
[Write the function.]

′ (x) = limh0f (x+h)-f (x)h
[Use the definition of '(x).]

= limh0(169(x+h)+12-169x+12h)
[Use the definition of (x).]

= 16limh0(9x+12)-(9(x+h)+12)h(9(x+h)+12) (9x+12)

= 16 limh0(- 9(9(x+h)+12) (9x+12))

= - 144(9x+12)2

′ (x) = - 144(9 x+12)2

10.
Calculate the derivative of ($l$) = $\frac{5l+8}{13l+17}$ from the definition of derivative.
 a. $\frac{19}{{\left(13l+17\right)}^{2}}$ b. $\frac{-19}{{\left(13l-17\right)}^{2}}$ c. $\frac{{\left(13l+17\right)}^{2}}{189}$ d. $\frac{189}{{\left(13l+17\right)}^{2}}$ e. $\frac{-19}{{\left(13l+17\right)}^{2}}$

#### Solution:

(l) = 5l+813l+17
[Write the function.]

′ (l) = limh0f (l+h) - f (l)h
[Use the definition of ′ (l).]

= limh0(5(l+h)+813(l+h)+17-5l+813l+17h)
[Use the definition of (l).]

= -19(13l+17)2
[Evaluate the limit.]

′ (l) = -19(13l+17)2