# Derivative with the Tangent Line Problem Worksheet

Derivative with the Tangent Line Problem Worksheet
• Page 1
1.
Find the slope of the curve $y$ = $\frac{x}{5x+8}$ at the point $x$ = 2.
 a. $\frac{8}{9}$ b. $\frac{4}{9}$ c. $\frac{1}{5}$ d. $\frac{7}{81}$ e. $\frac{2}{81}$

#### Solution:

y = x5x+8
[Equation of the curve.]

dydx = ddx (x5x+8)
[Find dydx.]

= (5x+8)(1)-x(5)(5x+8)2
[Use Quotient Rule.]

= 8(5x+8)2
[Simplify.]

(dydx)x=2 = 8(5(2)+8)2
[Find the slope of the curve at x = 2]

= 2 / 81
[Simplify.]

So, the slope of the curve at x = 2 is 2 / 81.

2.
Find the slope of the curve $y$ = 6$x$2 + 7$x$ at $x$ = 3.
 a. 36 b. 25 c. 43 d. 387 e. 75

#### Solution:

y = 6x2 + 7x
[Equation of the curve.]

dydx = ddx (6x2 + 7x)
[Find dydx.]

dydx = 12x + 7
[Use Sum Rule.]

(dydx)x = 3 = 12(3) + 7 = 43
[Find the slope of the curve at x = 3]

So, the slope of the curve at x = 3 is 43.

3.
Find the slope of the curve $y$ = 6$x$3 at $x$ = 2$a$.
 a. 48$a$4 b. 24$a$2 c. 96$a$4 d. 3$a$2 e. 72$a$2

#### Solution:

y = 6x3
[Equation of the curve.]

dydx = ddx (6x3)
[Find dydx.]

dydx = 18x2

(dydx)x = 2a = 72a2
[Find the slope of the curve at x = 2a]

So, the slope of the curve at x = 2a is 72a2.

4.
Find the slope of the curve $y$ = $\frac{6x+3}{3{x}^{2}+6x+4}$ at $x$ = 1.
 a. $\frac{42}{169}$ b. $\frac{9}{13}$ c. - $\frac{30}{169}$ d. $\frac{2}{3}$ e. - $\frac{132}{169}$

#### Solution:

y = 6x+33x2+6x+4
[Equation of the curve.]

dydx = ddx (6x+33x2+6x+4)
[Find dydx.]

dydx = (3x2+6x+4)(6)-(6x+3)(6x+6)(3x2+6x+4)2
[Use Quotient Rule.]

= - 18x2 - 18x+6(3x2+6x+4)2
[Simplify the numerator.]

(dydx)x = 1 = -18-18+6169
[Find the slope of the curve at x = 1]

= - 30 / 169
[Simplify.]

So, the slope of the curve at x = 1 is - 30 / 169.

5.
Find the slope of the curve $y$ = $\frac{9}{8x}$ at $x$ = 4.
 a. $\frac{1}{8}$ b. $\frac{9}{32}$ c. - $\frac{9}{128}$ d. 72

#### Solution:

y = 98x
[Equation of the curve.]

dydx = ddx (98x)
[Find dydx.]

dydx = 8x(0) - 9(8)64x2
[Use Quotient Rule.]

= - 7264x2
[Simplify the numerator.]

(dydx)x = 4 = - 7264(4)2 = - 9 / 128
[Find the slope of the curve at x = 4]

So, the slope of the curve at x = 4 is - 9 / 128.

6.
Find the slope of the curve $y$ = $\frac{7}{{x}^{2}}$ at $x$ = 4.
 a. $\frac{49}{32}$ b. $\frac{7}{32}$ c. $\frac{1}{16}$ d. - $\frac{7}{4}$ e. - $\frac{7}{32}$

#### Solution:

y = 7x2
[Equation of the curve.]

dydx = ddx (7x-2)
[Find dydx.]

dydx = - 14x- 3 = - 14x3
[Use Power Rule.]

(dydx)x = 4 = - 14(4)3 = - 7 / 32
[Find the slope of the curve at x = 4]

So, the slope of the curve at x = 4 is - 7 / 32.

7.
Find the slope of the curve $y$ = $e$2$x$ + 5 at $x$ = 3.
 a. $e$32 b. 2$e$11 c. 11$e$10 d. $e$11 e. $e$10

#### Solution:

y = e2x + 5
[Equation of the curve.]

dydx = ddx(e2x + 5)
[Find dydx.]

dydx = 2e2x + 5
[Use Chain Rule.]

(dydx)x = 3 = 2e11
[Find the slope of the curve at x = 3]

So, the slope of the curve at x = 3 is 2e11.

8.
Find the slope of the curve $y$ = $\frac{4x+5}{6x+7}$ at $x$ = 2.
 a. $\frac{13}{19}$ b. $\frac{2}{3}$ c. - $\frac{154}{19}$ d. - $\frac{2}{361}$ e. $\frac{154}{361}$

#### Solution:

y = 4x+56x+7
[Equation of the curve.]

dydx = ddx (4x+56x+7)
[Find dydx.]

dydx = (6x+7)(4)-(4x+5)(6)(6x+7)2
[Use Quotient Rule.]

= 24x+28-24x-30(6x+7)2
[Expand the numerator.]

= - 2(6x+7)2
[Simplify.]

(dydx)x = 2 = - 2(6(2)+7)2 = - 2361
[Find the slope of the curve at x = 2]

So, the slope of the curve at x = 2 is - 2 / 361.

9.
Find the slope of the curve $y$ = at $x$ = 3.
 a. 1 b. $\frac{18}{5}$ c. $\frac{1}{5}$ d. $\frac{1}{3}$ e. 5

#### Solution:

y = x+25
[Equation of the curve.]

dydx = 1 / 5ddx(x+2)
[Find dydx.]

= 15(1 + 0) = 15
[Use Sum Rule.]

(dydx)x = 3 = 15
[Find the slope of the curve at x = 3.]

So, the slope of the curve at x = 3 is 1 / 5.

10.
Find the slope of the curve $y$ = ln(3$x$ + 11) at $x$ = 3.
 a. ln (20) b. $\frac{1}{20}$ c. ln (38) d. $\frac{3}{20}$ e. $\frac{27}{20}$

#### Solution:

y = ln(3x + 11)
[Equation of the curve.]

dydx = ddx (ln(3x + 11))
[Find dydx.]

dydx = 33x+11
[Use Chain Rule.]

(dydx)x = 3 = 33(3)+11 = 3 / 20
[Find the slope of the curve at x = 3]

So, the slope of the curve at x = 3 is 3 / 20.