Brennan's Calculus

Derivatives:

100

d/dx (1/3 x^12 + 1/5x^20)

4x^11 + 4x^19

200

d/dx (3/7x^14 - 3x^8-2x^-3)

6x^13 - 24x^7 + 6x^-4

or

6x^13 - 24x^7 + 6/x^4

300

d/dx (1/8x^(8/3) - 5/2 x^(12/25))

1/3 x^(5/3) - 6/5 x^(-13/25)

Limits:

100

d/dx (x^2sinx)

x^2cosx + 2xsinx

200

d/dx((3x^5-1)(x^3+5x^2+2))

(3x^5-1)(3x^2+10x) + (15x^4)(x^3+5x^2+2)

or expanded form:

9x^7 + 30x^6-3x^2-10x + 15x^7 + 75x^6 + 30x^4 = 24x^7 + 105x^6 + 30x^4 - 3x^2 - 10x

300

d/dx((5x^5-x^7)(20x^2+3x^-7))

(5x^5-x^7)(40x-21x^-8)+(25x^4-7x^6)(20x^2+3x^-7)

or expanded form:

200x^6 - 105x^-3 -40x^8 +21x^-1 + 500x^6 + 75x^-3 -140x^8 -21x^-1 = 700x^6 - 30x^-3 -180x^8

Integrals

100

d/dx ((3x^2+9)/(x-5))

((x-5)(6x)-(3x^2+9)(1))/(x-5)^2 = (6x^2-30x-3x^2-9)/((x-5)^2) = (3x^2-30x-9)/((x-5)^2)

200

d/dx ((4sqrt(x))/(x^2-2))

((x^2-2)*2x^(-1/2) - 4x^(1/2)(2x))/(x^2-2)^2

=

(2x^(-1/2)(x^2-2) - 8x^(3/2))/((x^2-2)^2)

300

d/dx ((4sinx)/(2x+cosx))

((2x+cosx)(4cosx)-(4sinx)(2-sinx))/(2x+cosx)^2

Application

100

d/dx ((6x^2+7x)^4)

4(6x^2+7x)^3*(12x+7)

200

d/dx sqrt(x+3)

1/(2sqrt(x+3))

300

d/dx (3x+2)^7/(x-1)

((3x+2)^6*(18x-23))/(x-1)^2