Chapter 4 Test Review

Chain Rule

100

(x⁴ + 1)⁵

20x³ (x⁴ + 1)⁴

200

y = cos (5x⁴)

-20x³ sin (5x⁴)

300

(3x⁴ + 5)⁴ (5x³ + 1)

3x² (3x⁴ + 5)³ (95x⁴ + 16x + 25)

400

y = sin³ (5x⁴)

60x³ sin² (5x⁴) cos (5x⁴)

500

y=cos (sin (2x⁵))

=-10x⁴  sin (sin (2x⁵)) cos (2x⁵)

Implicit Differentiation

100

3x³=4y² + 2

(9x²)/(8y)

200

-5xy² + 5 = x

(-1 -5y²) / (10xy)

300

(5y² + 1)² =4x³

(3x²) / (25y³ + 5y)

400

(3y³ + 2)² =2x

1 / (27y⁵ + 18y²)

500

3x + 5 = sec (2y²)

3 / (4y sec (2y²) tan (2y²))

Inverse Functions/Trig

100

y=cos^-1(-5x⁴)

(20x³) / (1 - 25x⁸)^1/2

200

y=tan^-1(4x⁵)

(20x⁴) / (16x¹⁰ + 1)

300

y=csc^-1 (5x³)

-(15x²) / (|5x³| (25x⁶ - 1)^1/2

400

f(x) = x + 4, a=3

(f^-1)' (a) = 1

500

f(x) = 5x - 3, a = -2

(f^-1)' (a) = 1/5

Exp and Log Functions

100

y=ln (4x⁵)

5/x

200

y=2^{5x^4}

5x³ * 2^{5x^4 + 2} ln 2

300

y=log₃(4x³)

3/ (x ln 3)

400

y=5x^{x^3}

y(3x² ln x + x²)

500

y= ((4x³ + 3)⁴) / ((x +1)³ * (5x² - 2)⁵)

[((4x³ + 3)⁴) / ((x +1)³ * (5x² - 2)⁵) ] * [(48x²)/(4x³ + 3) - (3) / (x + 1) - (50x) / (5x² - 2)]

Wild Card

100

y= 2x² - 4x - 2 at (2, -2)

Find the equation of the normal line.

y= -(1/4) x - 3/2

200

y= -2  cos (2x) at (pi/4, 0)

Find the equation of the tangent line.

y = 4x - pi

300

y= -x³ + x²

Find the points where the tangent line is horizontal.

(0,0), (2/3, 4/27)

400

x² = 3y² + 2

Find (d²y)/(dx²).

(d² y)/(dx²) = (3y² -x²) / (9y³)

500

5x² -4y² =3

Find (d²y)/(dx²).

(d²y) / (dx²) = (20y² - 25x²) / (16y³)