Less9ProdQuotRulesTrigDeriv

Question 1

Prove that the derivative of the product is not the product of the derivatives with:

f(x) = x g(x) = x²

Answer 1

f’(x) = 1 g’(x) = 2x

The product of the derivatives is 2x

x*x² = x³ It’s derivative is 3x²

Question 2

What is the Product Rule formula?

Answer 2

[f(x)*g’(x)] + [g(x)*(f’(x)]

Question 3

Use the Product Rule to solve f(x) = x*x²

Answer 3

f(x) = x f’(x) = 1

g(x) = x² g’(x) = 2x;

(x*2x) + (x²*1) = 3x²

Question 4

Use the Product Rule to find the derivative of:

3x²*sinx

Answer 4

f(x) = 3x² f’(x) = 6x;

g(x) = sinx g’(x) = cosx;

(3x²*cosx) + (sinx*6x) =

3x(xcosx + 2sinx)

Question 5

What is the formula for the Quotient Rule of derivatives?

Answer 5

^d/_dx[f(x)/g(x)] =

^{[g(x)f’(x) - f(x)g’(x)]}/_g²(x)

g(x) not=0

Question 6

What is the derivative of (x³+cosx)/6?

Answer 6

= 1/6*(x³ + cosx)

Use the constant multiple rule

d/dx = 1/6*(3x² - sinx)

You don’t need the quotient rule.

Question 7

What is the derivative of tanx?

Answer 7

sin’(x)/cos’(x)

f(x)=sinx f’(x)=cosx

g(x)=cosx g’(x)=-sinx

^{(cosxcosx) - (sinx-sinx)}/_cos²x

cos²x + sin²x = 1 so

1/cos²x = sec²x

Question 8

How do you find the cotx on a TI-84?

Answer 8

The cotx is 1/tanx.

tan(1 radian) ≈ 1.557

1/1.557 ≈ .6421

Question 9

What is the derivative of cotx?

Answer 9

cotx = cosx/sinx

^{f(x) = cosx f’(x) = -sinx}

^{g(x) = sinx g’(x) = cosx}

^{g²(x) = sin²x}

[-sin²x - cos²x]/sin²x

-(sin²x + cos²x)/sin²x

^{[sin²x + cos²x = 1]}

-1/sin²x = -csc²x

^d/_dxcotx = -csc²x

Question 10

What is the derivative of secx?

Answer 10

secx = 1/cosx

^d/_dxsecx = secxtanx

Question 11

What is the derivative of cscx?

Answer 11

^d/_dxcscx = -cscxcotx

Question 12

What is ^d/_dx (3x - tanx)?

Answer 12

3 - sec²x

Question 13

What are higher orders of derivatives?

Answer 13

Derivatives of derivatives

Question 14

What are the first five orders of derivatives of f(x) = x⁴?

Answer 14

f’(x) = 4x³

f”(x) = 12x²

f”‘(x) = 24x

f”“(x) = 24

f””‘(x) = 0

all higher ones are 0

Question 15

How do you find the numerical derivative at any point on a graph on a TI-84?

Answer 15

Enter equation on y= screen → Graph → 2nd/Trace → Enter the letter x → x-value

You must go through this process each time for different x-values

Question 16

Give an example of a practical use of higher derivatives,

Answer 16

s(t) = position

s’(t) = instantaneous velocity

s”(t) = acceleration

Question 17

Find 4 levels of derivatives for f(x) = sinx

Answer 17

f(x) = sinx

f’(x) = cosx

f”(x) = -sinx

f”‘(x) = -cosx

f”“(x) = sinx

Question 18

What are the equations for a falling object?

Answer 18

s(t) = ½gt² + v₀t +s₀

_{g=gravitational constant, v0=initial velocity s0=initial position}

s’(t) = gt + v₀

s”(t) = v’(t) = a(t) acceleration

Question 19

What is a differential equation?

Answer 19

an equation that has derivatives in it.

Given the derivatives, what is the underlying equation?

Question 20

Where are the tangent lines horizontal with f(x) = x⁴ - 2x² + 3?

Answer 20

f’(x) = 4x³ - 4x

4x³ - 4x = 0 ^(horizontal)

4x³ = 4x

x³ = x

x = 0, 1, -1

(0,3); (1,2); (-1,2)

Question 21

What is the equation for the tangent line at the origin of

f(x) = ^16x/_{(x² + 16)}?

Answer 21

Quotient Rule:

f(x)= 16x f’(x)= 16

g(x)=x² + 16 g’(x)=2x

^{(x²+16)16)) - (16x2x)}/_(x²+16)²

No need to simplify. At x=0,

256/256 = 1

((y - 0) = 1(x - 0) :: y = x

Question 22

Use the Product Rule to find ^d/_dx (x² + 3)(x² - 4x)

Answer 22

^d/_dx (x² + 3)(x² - 4x)

f(x)=(x²+3) f’(x)=2x

g(x)=(x²-4x) g’(x)=(2x-4)

(x²+3)*(2x-4) + (x²-4x)(2x) =

(2x³-4x²+6x-12) + (2x³ - 8x²)

4x³-12x² + 6x-12

Question 23

Use the quotient rule to find the derivative of:

f(x) = ^x/_{(x² + 1)}

Answer 23

f(x) = ^x/_{(x² + 1)}

f(x)=x f’(x)=1

g(x)=x²+1 g’(x)=2x

^{((x²+1)1) - (x2x)}/_(x²+1)²

(1-x²)/(x+1)²

Question 24

Use the quotient rule to find the derivative of:

^sinx/_x²

Answer 24

f(x) = sinx/x²

f(x)=sinx f’(x)=cosx

g(x)=x² g’(x)=2x

^{x²cosx - sinx2x}/_x⁴

^{x²cosx - sinx2x}/_x⁴

^{(xcosx - 2sinx)}/_x³

Question 25

Find the equation of the tangent line to the graph

f(x) = (x³ + 4x - 1)(x - 2)

at (1, -4)

Answer 25

f(x)=x³+4x-1 f’(x)=3x²+4

g(x)=x-2 g’(x) = 1

(x-2)*(3x²+4) + (x³+4x-1)*1

(3x³-6x²+4x-8) + (x²+4x-1)

f(1) = -7 +4 = -3 = slope

(y - -4) = -3(x - 1)

y = -3x - 1

Question 26

Find the derivative of f(x) = 10/3x³ without the quotient rule.

Answer 26

= ¹⁰/₃ * x^-³

= ¹⁰/₃*-3x^-4

¹⁰/₃*-³/x⁴

^-10/x^-4

Question 27

Find the derivative of f(x) = x²tan(x)

Answer 27

f(x)=x² f’(x) = 2x

g(x)=tanx g’(x)=sec²x

x²*sec²x + tanx*2x

x(xsec²x + 2tanx)

Question 28

Find the derivative of f(x) = x²tan(x)

Answer 28

f(x)=x² f’(x) = 2x

g(x)=tanx g’(x) = sec²x

x²*sec²x + 2x*tanx

x(xsec²x + 2tanx)

Question 29

Find the derivative of ^secx/_x

Answer 29

f(x) = secx f’(x) = secxtanx

g(x)=x g’(x)= 1

^{x(secxtanx) - secx}/_x²

^{secx(xtanx - 1)}/_x²

Question 30

Find the second derivative of f(x) = 4x^3/2

Answer 30

f’(x) = ³/₂*4x^½= 6x^½

f”(x) = 3x^-½ = ³/_√x

Question 31

Find the second derivative of f(x) = x/(x-1)

Answer 31

f(x) = x f’(x) = 1

g(x)= x-1 g’(x)=1

(x - 1) - x/(x-1)² = -1/(x-1)²

f’(x) = -(x-1)^-2

f”(x) = 2(x-1)^-3 = 2/(x-1)³

Question 32

What is a differential equation?

Answer 32

An equation that has derivatives in it

Question 33

Show that y = sinx satisfies y” + y = 0

Answer 33

f(x) = sinx

f’(x) = cosx f”(x) = -sinx

• sinx + sinx = 0

Question 34

Show that y = cosx also satisfies y” + y = 0

Answer 34

f(x) = cosx

f’(x) = -sinx f”(x) = -cosx

-cosx + cosx = 0