The Chain Rule Extravaganza

Derivatives

100

y=3^x

ln3 x 3x

200

y=ln(5x⁷)³

21/x

300

y=cos³(4x)

-12sin(4x)cos²(4x)

400

y=e^3x(2)-7x

^{*A parentheses in an exponents is to show it’s an exponent on an exponent.}

6e^3x(2)-7xx-7e^3x(2)-7x

500

y=arccos(x⁷)

Hint: the answer includes a square root, however, it will look like a parentheses with a number below it because of software issues.

-7x⁶/(1-x¹⁴)₂

Definitions

100

What is y’’?

Double Derivative

200

What is a monotonic function?

(Your answer doesn’t have to exact).

A function which is either entirely nonincreasing or nondecreasing.

300

What is implicit relations?

(Your answer doesn’t have to exact).

Depended variable isn’t isolated on one side.

400

Name one of the definitions of continuity.

1) f(x) is defined.

2) lim f(x) exist.

3) lim f(x) = f(a).

500

What is a higher-order derivative?

Hint: It’s the opposite of Definitions 200. :)

The derivatives other than the first derivative.

Double Derivatives

100

Find f’’(x)

6x⁴ + 7x³ - 18x² - 13x + 6

f”(x) = 72x² + 42x - 36

200

Find y’’’’

y = x⁵ + 10x³

y’’’’ = 120x

300

Find d²y/dx² for

sin(y) = x + y

sin(y)/(cos - 1)³

400

Find y’’’

y = e^-5x + 8ln(2x⁴)

y’’’ = -125e^-5x - 192x^-4

500

Find f’’(x)

f(x) = ln(9x + 2)

f’’(x) = -81/(9x + 2)²

Implicit Differentiation

100

x/y³ = 1

y’ = y^-3/3xy^-4

or

y’ = y/3x

200

e^x - sin(y) = x

y’ = (1 - e^x)/(-cos(y))

or

y’ = (e^x - 1)sec(y)

300

1/x + 1/y = -1/12

y’ = -y²/x²

400

7y² + sin(3x) = 12 - y⁴

y’ = (-3cos(3x))/(14y + 4y³)

500

Evaluate at (0,3)

y²e^2x = 3y + x²

-6

Random

100

find d²y/dx²

y = y² + x

2/(1-2y)³

200

Find the derivative

x² + 2y⁵ = 10xy

y’ = (5y - x)/(5y⁴ - 5x)

300

Find the equation of the tangent line for 

x² + 7y² = 8y³ at (-6, 2)

y - 2 = (-13/7)(x + 6)

400

find y’
y = tan(x)/csc(x)

y’ = (csc(x)sec²(x) + tan(x)csc(x)cot(x))/csc²(x)

or

y’ = (sec²(x) + 1)/csc(x)

500

Find y’

y = tan^-1(4x²)

y’ = (1/(1 + (4x²)²))4