Less10ChainRule

Question 1

When is the chain rule helpful?

Answer 1

With composite functions. A composite function is a function within another function.

Question 2

What is the derivative of f(x)=(x²+1)³?

Answer 2

The inside function is x²+1

The outside function is inside³

f’(x)=3(x²+1)²*2x =6x(x²+1)²

It is the derivative of the outside * derivative of the inside.

Question 3

What is the definition of the chain rule?

Answer 3

It is calculated by multiplying the derivative of the outside function by the derivative of the inside function.

f’(g(x))*g’(x)

Question 4

What is the derivative of

f(x) = (3x-2x²)⁵

Answer 4

inside = 3x-2x²

outside= ()⁵

f’(x)= 5(3x-2x²)⁴*(-4x+3)

Question 5

What is the derivative of √(3x+1)?

Answer 5

inside = 3x+1

outside = √() or ()^½

½ (3x+1)^-½*3

=³/_2√(3x+1)

Question 6

What is the General Power Rule?

Answer 6

It is a generalized form of the Power Rule

and

a specialized form of the Chain Rule:

^d/_dx[u(x)]ⁿ = nu^n-1*u’

where u is the inside function

Question 7

What is the derivative of

f(x) = sin(6x)?

Answer 7

inside = 6x outside = sin()

cos(6x)*6

Question 8

What is the derivative of

f(x) = x²*√(5-x²) at point (1,2)?

Answer 8

We need the chain rule and the product rule.

f(x) = x² f’(x) = 2x

g(x) = √(5-x²)

g’(x) = [½(5-x²)^-½*(-2x)]

x²*½(5-x²)^-½*(-2x) + √(5-x²)(2x) Solve at (1,2)

^{You don’t need to simplify if you are going to plug in a point.}

1*½(4)^-½(-2) + 4^½*2=

-½ + 4 = ⁷/₂

Question 9

What is the derivative of

f(x) = sin³(4x)

Answer 9

^{This requires a double application of the chain rule.}

^{Also, sin³(4x = (sin4x)³}

3(sin4x)²*^d/_dx(sin(4x))

3(sin4x)²*(cos4x)*4

=12sin²4x*cos4x

Question 10

What is the derivative of

f(x) = sin²x + cos²x ?

Answer 10

f’(x) sin²x = 2sinx*cosx

f’(x) cos²x = 2cosx*-sinx

2(sinx)(cosx) -2(cosx)(sinx) = 0

^{sin²x + cos²x = 1 (trig ident)}

^{The graph of this equation is y=1, whose slope is 0}

Question 11

What is the derivative of tan²x - sec²x ?

Answer 11

f(x) = tan²x f’(x) = 2(tanx)(sec²x)

g(x) = sec²(x) g’(x)= 2(secx)(secx)(tanx)

2(tanx)(secx)(secx) - 2(secx)(secx)(tanx) = 0

^{Trig identity: tan²x + 1 = sec²x}

^{The equation of tan²x - sec²x is y= -1}

Question 12

What is the second derivative of f(x) = ¹/_(2x-6) ?

Answer 12

f(x) = (2x-6)^-1

f’(x) = -(2x-6)^-2*2 = -2(2x-6)^-2

f”(x) = 4(2x-6)^-3*2 =

⁸/_(2x-6)³

Question 13

You have a 15 cm long pendulum moving at Θ=.2cos8t.

Calculate how far out it goes

^{Θ is the angle of the pendulum.}

Answer 13

At t=0, cos(0) = 1 ::

Θ=.2 radians

^{This is the maximum displacement because cos can never be greater than 1.}

^{At t=π/16, Θ=π/2 :: cos=0 vertical}

Question 14

Calculate the rate of change of the angle (velocity).

Answer 14

8t=inside cos8t=outside

dΘ/dt = .2(-sin8t)*8 = -1.6(sin8t) rad/sec

at t=0, velocity=0 ^{(the pendulum starts out at .2 radians when it is dropped.)}

Question 15

Find the points on f(x)=2cosx + sin2x on the interval (0,2π) which have horizontal tangents.

Answer 15

f’(x) = -2sinx + 2cos2x

Set = 0. sinx - cos2x = 0

^{Trig identity: cos2x = cos²x - sin²x = 2cos²x - 1 = 1 - 2sin²x}

sinx - (1-2sin²x)=2sin²x + sinx -1 = 0

=(2sinx-1)(sinx+1) =0

sinx = ½ (x=^π/₆ or ^5π/₆)

or sinx = -1 (x=^3π/₂)

Question 16

What is the derivative of

f(x) = (3x + 5)⁵⁰

Answer 16

f’(x) = 50(3x+5)⁴⁹(3) = 150(3x+5)⁴⁹

Question 17

How can you tell if it is a composite function?

Answer 17

(1) y = √x is not because the argument is just x
(2) y = √x*sinx is not because it’s just a product function

Question 18

How do you find the “inside” of a composite function?

Answer 18

Look for parentheses or something that can be put in parentheses.

sin²x = (sinx)²

Question 19

How do you rewrite

y = sin³(5x² - 4x)?

Answer 19

(sin(5x² - 4x))³

^{This has 3 levels of composition.}

^{3(sin(5x² - 4z))²*(sin(5x² - 4x)’}

^{=3(sin(5x² - 4z))²*(cos(5x² - 4x)*(10x - 4)}

Question 20

Find the derivative of

f(x) = (4x - 1)³

Answer 20

3(4x - 1)²* 4 = 12(4x - 1)²

Question 21

Find the derivative of

f(t) = √(5 - t)

Answer 21

f(t) = (5 - t)^½

f’(t) = ½(5 - t)^-½*(-1)

f’(t) = -½(5 - t)^-½

= ^-1 /_{2√(5 - t)}

Question 22

What is the derivative of

f(t) = cos4x ?

Answer 22

f(t) = cos(4x)

f’(t) = (-sin(4x))*4

Question 23

Find the derivative of

f(t) = 5tan3x

Answer 23

f(t) = 5tan(3x)

f’(t) = 5*sec²3x*3 = 15(sec²3x)

Question 24

Find the equation of the line tangent to

f(x) = (9 - x²)^⅔ at (1,4)

Answer 24

f’(x) = ⅔(9-x²)^-⅓*(-2x)

f’(t) = -4x/3(9-x²)^⅓

f’(1) = -4/3√8 = -4/6 = -⅔

The equation of the tangent line:

(y - 4) = -⅔(x - 1)

y = -⅔x + 4 ⅔

Question 25

Find the derivative of:

f(x) = 5/(x³ - 2) at (-2, -½)

Answer 25

f(x) = 5*(x³ - 2)^-1

f’(x) = 5*-(x³ - 2)^-2(3x²)

f’(x) = -15x²/(x³-2)²

f’(-2) = -60/100 = -3/5

Question 26

Find the second derivative of:

f(x) = ⁴/_{(x + 2)³}

Answer 26

f(x) = 4*(x + 2)^-3

f’(x) = 4*-3(x + 2)^-4(1)

f”(x) = 4*12*(x + 2)^-5(1)

= 48/(x + 2)⁵

Question 27

Find the derivative of:

f(x) = tan²x - sec²x

Answer 27

f(x) = (tan(x))² - (sec(x))²

f’(x) = 2tanxsec²x - 2secx*secxtanx

=2tansec²x - 2tansec²x = 0

^{We know from the original problem that tan²x + 1 = sec²x (trig identity)}

^{So the derivatve of 1 is 0}