Derivative Rules

The Big 4

100

d/dx

(x⁴ + 3x³ - 7x² +3)

= 4x³ + 9x² -14x

200

d/dx

(6x² + 7x)² 

= 2(6x² +7x)(12x + 7)

= 144x³ + 252x² + 98x

300

d/dx

(x²sin(x))

= 2xsin(x) + x²cos(x) 

400

d/dx

[xsin(3x)]

= sin(3x) + 3xcos(3x)

500

d/dx

[(3/2)x² + 4x] / [3x⁴ + 7x²]

= [(3x⁴ + 7x²)(3x + 4) - ((3/2)x² + 4x)(12x³ + 14x)] / [(3x⁴ -7x²)²]

= (-9x³ - 36x² - 28) / (9x⁶ +42x⁴ + 49x²)

Logarithmic and Exponential

100

d/dx

(e^2x)

= 2e^2x

200

d/dx

(log₆x)

= 1/(xln(6))

300

d/dx

[ln(7x² + 2)]

= (14x) / (7x² + 2)

400

d/dx

[e^3xln(x²)]

= 3e^3xln(x²) + (2e^3x)/x

500

d/dx

(4^x / ln(x))

= [ln(x)4^xln(4) - (4^x / x)] / (ln(x))²

Trigonometry and Inverse

100

d/dx

[cos(x)]

= -sin(x)

200

d/dx

[sec(x)]

= sec(x)tan(x)

300

d/dx

[sin^-1(x)]

= 1 / (1 - x²)^1/2

400

d/dx

[cos(x)tan(x) - 3]

= -sin(x)tan(x) + sec(x)

= -sin(x)tan(x) + cos(x)sec²(x)

500

d/dx

[tan^-1(sin(x))]

= cos(x) / (1 + sin²(x))

Wild Card

100

d/dx

(4^ln(6))

= 0

200

d/dx

(e^2cos(x))

= -2sin(x)e^2cos(x)

300

d/dx

[3xln(4x²)]

= 6 + 3ln(4x²)

400

g is the inverse function of f. Find g'(1). 

Points on f(x): (1,-5) (2,-9)

Points on f'(x): (1,-2) (2,1)

Points on g(x): (1,2) (2,4)

= 1

500

d/dx

[e^3x / ln(4x)]

= (ln(4x)3e^3x - e^3x/x) / (ln(4x))²