Calculus Concepts

Basic Differentiation

d/dx sin(u) = cos(u) du/dx

Example:
d/dx sin(3x)
u = 3x, so du/dx = 3
Therefore the answer is cos(3x) * 3 or
3 cos(3x)

d/dx cos(u) = -sin(u) du/dx

Example:
d/dx cos(x^2)
u = x^2, so du/dx = 2x
Therefore the answer is -sin(x^2) * 2x or
-2x sin(x^2)

d/dx tan(u) = sec^2(u) du/dx

Example:
d/dx tan(5x)
u = 5x, so du/dx = 5
Therefore the answer is sec^2(5x) * 5 or
5 sec^2(5x)

d/dx cot(u) = -csc^2(u) du/dx

Example:
d/dx cot(3x^3)
u = 3x^3, so du/dx = 9x^2
Therefore the answer is -csc^2(3x^3) * 9x^2 or
-9x^2 csc^2(3x^3)

d/dx sec(u) = sec(u)tan(u) du/dx

Example:
d/dx sec(7x)
u = 7x, so du/dx = 7
Therefore the answer is sec(7x)tan(7x) * 7 or
7 sec(7x)tan(7x)

d/dx csc(u) = -csc(u)cot(u) du/dx

Example:
d/dx csc(18x)
u = 18x, so du/dx = 18
Therefore the answer is -csc(18x)cot(18x) * 18 or
-18 csc(18x)cot(18x)

d/dx e^u = e^u du/dx

Example:

d/dx e^(9x)

u = 9x, so du/dx = 9

Therefore the answer is e^(9x) * 9 or

9 e^(9x)

d/dx a^u = ln(a) * a^u du/dx

Example:

d/dx 3^x

u = x, so du/dx = 1

Therefore the answer is ln(3)* 3^x * 1 or

ln(3) * 3^x

d/dx ln(u) = (1/u) du/dx

Example:

d/dx ln(x^3)

u = x^3, so du/dx = 3x^2

Therefore the answer is [1/(x^3)] * 3x^2 or

(3x^2)/(x^3) = 3/x

d/dx ln abs[u] = (1/u) du/dx

Example:

d/dx ln abs[15x]

u = 15x, so du/dx = 15

Therefore the answer is [1/(15x)] * 15 or

15/(15x) = 1/x

d/dx loga(u) = [1/ln(a)] * [1/u] du/dx

Example:

d/dx log3(4x)

u = 4x, so du/dx = 4

Therefore the answer is [1/ln(3)] [1/(4x)] * 4 or

4 [1/ln(3)] [1/(4x)] = [1/ln(3)] [1/x]

Advanced Differentiation

d/dx arcsin(u/a) =

1 / √(a^2 – u^2) du/dx

d/dx arccos(u/a) =

-1 / √(a^2 – u^2) du/dx

d/dx arctan(u/a) =

a / (a^2 + u^2) du/dx

d/dx arccot(u/a) =

- a / (a^2 + u^2) du/dx

d/dx arcsec(u/a) =

a / [u / u√(u^2 – a^2)] du/dx

d/dx arccsc(u/a) =

- a / [u / u√(u^2 – a^2)] du/dx

Differentiation

Integration

Rules

Logic Problems!